Robust finite-time stabilization for positive delayed semi-Markovian switching systems

doi:10.1016/j.amc.2018.12.069

Applied Mathematics and Computation

Volume 351, 15 June 2019, Pages 139-152

https://doi.org/10.1016/j.amc.2018.12.069 Get rights and content

Abstract

Robust finite-time stabilization is discussed for positive semi-Markovian switching systems (S-MSSs), in which semi-Markovian process, time-varying delay, external disturbance, and transient performance in finite-time level are all considered in a unified framework. In the system under consideration, finite-time problem can describe transient performance of practical control process. Firstly, some finite-time boundedness and $L_{1}$ finite-time boundedness criteria for positive delayed S-MSSs are proposed by constructing the stochastic semi-Markovian Lyapunov–Krasovskii functional with mode-dependent integral term. Then, a developed $L_{1}$ finite-time feedback controller design method is presented to reduce some constraints of input matrices, which guarantees the resulting closed-loop system achieves positivity, finite-time boundedness, and has a prescribed $L_{1}$ noise attenuation performance index in a novel standard linear programming. Finally, a practical example is illustrated to validate the proposed results by applying a virus mutation treatment model.

Introduction

Positive systems (also referred to as nonnegative systems) are a special class of dynamical systems whose input variables, output variables, and state variables need to satisfy the requirements of nonnegative properties [1], [2]. This kind of systems find wide applications in vehicle formation control, air traffic flow control, economics, sociology, ecology, and engineering, and have attracted much attention over the past several years (see e.g., [3], [4], [5], [6], [7], [8], [9], [10]). For example, in order to investigate a minimal positive realization for positive linear systems, the authors have used downsampling to obtain minimal positive realizations matching decimated sequences of Markovian coefficients of the impulse response [3]. In the literature [5], by the use of linear co-positive Lyapunov functions, results for the synthesis of stabilizing, guaranteed performance and optimal control laws for switched linear systems have been presented to be applied to a simplified human immunodeficiency viral mutation model. Necessary and sufficient conditions for synthesis of state-feedback controllers have been solved in terms of linear programming problem including the requirement of positiveness of the controller [8].

On the other hand, there always exists time delay in measurement process of system variable, large inertia element of system equipment, incubation period and genetic problems of infectious diseases, slow thermal reaction process, and signal communication of network control systems, which has been regarded as a key factor to destroy system stability [11]. Over the past decades, a quantity of results have been reported for positive delayed systems (see e.g., [12], [13], [14]). For example, the ℓ_∞-gain (respectively, $L_{\infty}$ -gain) analysis problem for discrete-time (respectively, continuous-time) positive linear systems with unbounded time-varying delays have been addressed by virtue of the monotonicity of an auxiliary system and the corresponding delay-free system in [12]. In the literature [13], under unbounded delay, several stability results have been established by constructing a sequence of functions that are positive, monotonically decreasing, and convergent to zero as time tends to infinity.

Additionally, in many practical applications, it is difficult to avoid some unpredictable structural changes. These abrupt changes often have a big impact on the original work environment and operating mechanism, and even destabilize the system. And they can be usually characterized by Markovian switching systems (MSSs). As a special kind of hybrid dynamics [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], MSSs have some advantages in modeling many practical systems, such as economic systems, flight systems, power systems, communication systems, fault-tolerant systems, manufacturing systems, circuits systems, and networked control systems. It is well known that dynamic characteristics of stochastic jump systems are restricted by probability distribution function of sojourn-time. Compared with MSSs whose sojourn-time follows unique memoryless exponential distribution [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], S-MSSs with the sojourn-time of semi-Markovian process obeying non-exponential distribution can be well suited to model more complex systems, in which the transition rate reflects the time-varying property. In fact, S-MSSs have many applications in describing dynamical systems, such as particular control problem in accelerator physics known as hospital planning, performance analysis of multiple-bus systems, and credit risk assessment. Therefore, the stability issue and control synthesis for S-MSSs have been extensively studied during the past years (see e.g., [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60]). To mention a few, the problem of stochastic stability for linear systems with jump parameters being semi-Markovian rather than full Markovian has been further investigated by using the comparison principle [49]. In the literature [50], numerically solvable sufficient conditions for stochastic stability and robust state feedback controller design of MSSs have been established by incorporating the upper and lower bounds of the transition rate and applying a new partition scheme. The problems of stability and stabilization for a class of discrete-time semi-Markovian switching linear systems have been investigated by introducing the discrete-time semi-Markovian kernel method [52]. Necessary and sufficient conditions for stochastic stability and $L_{1}$ state-feedback controller of S-MSSs [55] have been established in standard linear programming. Based on the T-S fuzzy model approach and semi-Markovian Kernel concept, several criteria ensuring σ-error mean square stability of the underlying closed-loop system have been established in [58].

Furthermore, it is noted that many existing results on analysis and synthesis for dynamic systems are based on Lyapunov stability in an infinite-time level. However, many practical processes are often concerned with dynamic behaviors in finite-time region, such as the trajectory problem of control aircraft transferred from one point to another point in a particular time zone, the temperature and pressure keeping in a certain range over a fixed-time period, etc. This special requirement is usually described as finite-time stability, which means that the state responses don’t exceed a certain threshold over a fixed-time level. Recently, finite-time theory is widely used in communication systems, aircraft control, multi-agent coordination control, network control system, and has received an extensive attention in different fields of research interests [6], [15], [16], [17], [18], [35], [61]. For instance, a sufficient condition has been established for the finite-time bounded $H_{\infty}$ performance analysis of the closed-loop fuzzy MSSs system with fully considering the asynchronous premises [35]. An adaptive fuzzy controller has been constructed to address the finite-time tracking control problem for a class of strict feedback nonlinear systems by backstepping design with a tan-type barrier Lyapunov function [61].

This paper will deal with the issue of $L_{1}$ finite-time stabilization for positive S-MSSs with time-varying delay. It is necessary to point out the differences between the present work and the existing relative works [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60]. First, compared with the existing literatures [40], [41], [42], [43], [44], [45], [46], [47], [48], the stochastic switching law obeys the Markovian process and limits its applications. Second, the considered system in [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60] is a general system without taking positive constraint condition into account. Third, the literatures [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60] mainly investigated the stochastic jump systems in an infinite-time level while the model in this paper is considered in a finite-time level. The problem of $L_{1}$ finite-time stabilization for positive S-MSSs with delay is full of challenging and open all the time. More specifically, compared with positive MSSs with sojourn-time-independent transition rate, it is very difficult to handle positive S-MSSs with sojourn-time-dependent transition rate. Moreover, one needs to consider not only finite-time stability of dynamic systems governed by the semi-Markovian process, but also the constrained positivity. In addition, the dynamical behaviors are affected by many complicated factors including semi-Markovian process, time delay, finite-time level, disturbance input, and positivity, which motivate our current research work. The main contributions are highlighted as follows: (i) By the use of stochastic semi-Markovian Lyapunov–Krasovskii functional, finite-time boundedness criteria for positive delayed S-MSSs are given; (ii) Furthermore, sufficient conditions for a prescribed $L_{1}$ noise attenuation performance index in finite-time level are presented; (iii) By applying the linear programming optimization toolbox, a feedback controller depending on the gain matrix decomposition method and the bounds of the time-varying transition rate matrix is designed such that the resulting closed-loop system achieves positivity, finite-time boundedness, and has a prescribed $L_{1}$ noise attenuation performance index; (iv) Additionally, since the quantity of the virus mutation treatment model matches the positive condition in practical operating process, representing this practical system as positive S-MSSs with delay illustrates the effectiveness of the main algorithms.

Notations. $A \geq \geq 0$ ( ≤  ≤ 0,  >  > 0,  <  < 0) means that $A$ is nonnegative (nonpositive, positive, negative); 1-norm of ||x||₁ stands for ${∥ x ∥}_{1} = \sum_{k = 1}^{n} | x_{k} |,$ where x_k is the kth element of $x \in R^{n}$ ; $1_{n}$ denotes all-ones vector in $R^{n}$ . $1_{n}^{r}$ means that rth element of the vector $1_{n}$ is one. $E {\cdot}$ stands for mathematical expectation. iff means if and only if.

Section snippets

Problem statement and preliminaries

The positive systems with semi-Markovian process and delay are given as $\begin{matrix} \dot{x} (t) = A (ν_{t}) x (t) + A_{d} (ν_{t}) x (t - λ (t)) + B (ν_{t}) u (t) + H (ν_{t}) v (t), \\ y (t) = C (ν_{t}) x (t) + D (ν_{t}) v (t), \\ x (t_{0} + s) = φ (s), \forall s \in [- λ, 0], \end{matrix}$ where $x (t) \in R^{n}$ is the state vector, $u (t) \in R^{m}$ is the input vector, $w (t) \in R^{l}$ is the noise vector, and $y (t) \in R^{s}$ is the output vector. λ(t) is given as 0 < λ(t) ≤ λ, $\dot{λ} (t) \leq h_{1} < 1$ . {ν_t, t ≥ 0} stands for the semi-Markovian process in $Φ = {1, 2, \dots, N}$ with probability transitions $\begin{matrix} P r {ν_{t + \bar{Δ}} = β | ν_{t} = α} = {\begin{matrix} ρ_{α β} (h) \bar{Δ} + o (\bar{Δ}), & α \neq β, \\ 1 + ρ_{α α} (h) \bar{Δ} + o (\bar{Δ}), & α = β, \end{matrix} \end{matrix}$ where h

Finite-time boundedness analysis

Theorem 1

For given $\bar{T},$ $ρ \in R_{+},$ and d₁, $d_{2} \in R_{+}^{n}$ (d₁ >  > d₂ >  > 0), if there exist η_α, δ_1α, δ_2α, δ₁, $δ_{2} \in R_{+}^{n},$ ∀α ∈ Φ, and ι₁, ι₂, ι₃, ι₄, ι₅, $ι_{6} \in R_{+},$ such that $\begin{matrix} e^{ρ T} (ι_{2} + λ ι_{3} + λ ι_{4} + \frac{1}{2} λ^{2} ι_{5} + \frac{1}{2} λ^{2} ι_{6} + (1 - e^{- ρ \bar{T}}) σ \tilde{v}) < < ρ ι_{1}, \end{matrix}$ $\begin{matrix} ι_{1} d_{2} < < η_{α} < < ι_{2} d_{1}, δ_{1 α} < < ι_{3} d_{1}, δ_{2 α} < < ι_{4} d_{1}, δ_{1} < < ι_{5} d_{1}, δ_{2} < < ι_{6} d_{1}, \end{matrix}$ $\begin{matrix} {\bar{A}}_{α}^{T} η_{α} + δ_{1 α} + δ_{2 α} + λ δ_{1} + λ δ_{2} + \sum_{β = 1}^{N} ρ_{α β} (h) η_{β} - ρ η_{α} < < 0, \end{matrix}$ $\begin{matrix} A_{d α}^{T} η_{α} - (1 - h_{1}) δ_{1 α} < < 0, \end{matrix}$ $\begin{matrix} \sum_{β = 1}^{N} ρ_{α β} (h) δ_{1 β} \leq \leq δ_{1}, \sum_{β = 1}^{N} ρ_{α β} (h) δ_{2 β} \leq \leq δ_{2}, \end{matrix}$ then system (4) is finite-time bounded w.r.t. (d₁, d₂, $\bar{T},$ σ), where $\tilde{v} = \max_{z, α \in Z \times Φ} {{\tilde{v}}_{α q}},$ $Z = {1, 2, \dots, z},$ and ${\tilde{v}}_{α q}$ stands for the qth element of ${\tilde{v}}_{α} = H_{α}^{T} d_{1} ι$

$L_{1}$ finite-time boundedness analysis

Theorem 2

For given $\bar{T},$ ρ, $γ \in R_{+},$ and d₁, $d_{2} \in R_{+}^{n}$ (d₁ >  > d₂ >  > 0), if there exist η_α, δ_1α, δ_2α, δ₁, $δ_{2} \in R_{+}^{n},$ ∀α ∈ Φ, and positive constants ι₁, ι₂, ι₃, ι₄, ι₅, $ι_{6} \in R_{+},$ such that (6), (8), (9), and $\begin{matrix} e^{ρ T} (ι_{2} + λ ι_{3} + λ ι_{4} + \frac{1}{2} λ^{2} ι_{5} + \frac{1}{2} λ^{2} ι_{6} + (1 - e^{- ρ \bar{T}}) σ γ) < < ρ ι_{1}, \end{matrix}$ $\begin{matrix} {\bar{A}}_{α}^{T} η_{α} + δ_{1 α} + δ_{2 α} + λ δ_{1} + λ δ_{2} + \sum_{β = 1}^{N} ρ_{α β} (h) η_{β} - ρ η_{α} + C_{α}^{T} 1_{s} < < 0, \end{matrix}$ $\begin{matrix} H_{α}^{T} η_{α} + D_{α}^{T} 1_{s} - γ 1_{l} < < 0, \end{matrix}$ then system (4) is finite-time bounded with a prescribed $L_{1}$ noise attenuation performance index w.r.t. (d₁, d₂, $\bar{T},$ σ).

Proof

It is noted that (7) holds if (27) is true. Letting ${\tilde{v}}_{α} = γ$ in (5), we can get

Controller design

Theorem 3

Assume that $B_{α} \geq \geq 0$ . For given $\bar{T},$ ρ, $γ \in R_{+},$ and d₁, $d_{2} \in R_{+}^{n}$ (d₁ >  > d₂ >  > 0), if there exist η_α, δ_1α, δ_2α, δ₁, $δ_{2} \in R_{+}^{n},$ κ_pα, $κ_{α} \in R^{n},$ ∀α ∈ Φ, and ι₁, ι₂, ι₃, ι₄, ι₅, ι₆, $ɛ_{α} \in R_{+},$ such that (6), (8), (27), (29), and $\begin{matrix} A_{α}^{T} η_{α} + κ_{α} + δ_{1 α} + δ_{2 α} + λ δ_{1} + λ δ_{2} + \sum_{β = 1}^{N} {\underset{̲}{ρ}}_{α β} η_{β} - ρ η_{α} + C_{α}^{T} 1_{s} < < 0, \end{matrix}$ $\begin{matrix} A_{α}^{T} η_{α} + κ_{α} + δ_{1 α} + δ_{2 α} + λ δ_{1} + λ δ_{2} + \sum_{β = 1}^{N} {\bar{ρ}}_{α β} η_{β} - ρ η_{α} + C_{α}^{T} 1_{s} < < 0, \end{matrix}$ $\begin{matrix} \sum_{β = 1}^{N} {\underset{̲}{ρ}}_{α β} δ_{1 β} \leq \leq δ_{1}, \sum_{β = 1}^{N} {\bar{ρ}}_{α β} δ_{1 β} \leq \leq δ_{1}, \sum_{β = 1}^{N} {\underset{̲}{ρ}}_{α β} δ_{2 β} \leq \leq δ_{2}, \sum_{β = 1}^{N} {\bar{ρ}}_{α β} δ_{2 β} \leq \leq δ_{2}, \end{matrix}$ $\begin{matrix} A_{α} 1_{m}^{T} B_{α}^{T} η_{α} + B_{α} Σ_{p = 1}^{m} 1_{m}^{p} κ_{p α}^{T} + ɛ_{α} I \geq \geq 0, \end{matrix}$ $\begin{matrix} κ_{p α} \leq \leq κ_{α}, \end{matrix}$ then system (4) is positive and finite-time bounded with a prescribed $L_{1}$ noise

Case study

Consider the virus mutation treatment model [5] described by $\begin{matrix} \dot{x} (t) = & (R (ν_{t}) - δ I + ɛ M) x (t) + A_{d} (ν_{t}) x (t - τ (t)) \\ + B (ν_{t}) u (t) + G (ν_{t}) v (t), \\ y (t) = & C (ν_{t}) x (t) + D (ν_{t}) v (t), \end{matrix}$ where x(t), u(t), v(t), and y(t) are two different viral genotypes, control input, disturbance input, and control output, respectively; {ν_t, t ≥ 0}, ε, δ, $M = [M_{m n}],$ and $M_{m n} \in {0, 1}$ are semi-Markovian process in $Φ = {1, 2},$ mutation rate, death or decay rate, system matrices, and genetic connections between genotypes with the parameters given as: $\begin{matrix} R_{1} = & [\begin{matrix} 0.05 & 0.8 \\ 0.7 \end{matrix} \end{matrix}$

Conclusions

In this paper, we have investigated the robust finite-time stabilization for positive semi-Markovian switching systems with delay and external disturbance. Sufficient criteria are proposed to ensure that the resulting closed-loop system achieves positivity, finite-time boundedness with a prescribed $L_{1}$ -gain performance index. Furthermore, by using the gain matrix decomposition method, sufficient criteria for the aimed finite-time feedback controller design method are constructed in the strict LP

Acknowledgments

This work is supported by National Natural Science Foundation of China (61703231, 61773235, and 61703249), Natural Science Foundation of Shandong (ZR2017QF001, ZR2017PF001, and ZR2017MF063), Chinese Postdoctoral Science Foundation (2017M612235 and 2018T110670), Taishan Scholar Project (TSQN20161033), Shandong Province Higher Educational Science and Technology Program (J17KA079), and Excellent Experiment Project of Qufu Normal University (jp201728).

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Robust finite-time stabilization for positive delayed semi-Markovian switching systems

Abstract

Introduction

Section snippets

Problem statement and preliminaries

Finite-time boundedness analysis

L1 finite-time boundedness analysis

Controller design

Case study

Conclusions

Acknowledgments

Automatica

Appl. Math. Comput.

J. Frankl. Inst.

J. Frankl. Inst.

Inf. Sci.

J. Frankl. Inst.

Automatica

Fuzzy Sets Syst.

Neurocomputing

Automatica

Nonlinear Anal. Hybrid Syst.

Appl. Math. Comput.

Nonlinear Anal. Hybrid Syst.

Automatica

Nonlinear Anal. Hybrid Syst.

Automatica

J. Frankl. Inst.

ISA Trans.

Nonlinear Anal. Hybrid Syst.

Automatica

Positive Linear Systems: Theory and Applications

Positive 1D and 2D Systems

Reset stabilisation of positive linear systems

Int. J. Syst. Sci.

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