Research Article Neural Network Based Finite-Time
Transcript Of Research Article Neural Network Based Finite-Time
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 359265, 8 pages http://dx.doi.org/10.1155/2013/359265
Research Article Neural Network Based Finite-Time Stabilization for Discrete-Time Markov Jump Nonlinear Systems with Time Delays
Fei Chen,1 Fei Liu,1 and Hamid Reza Karimi2
1 Key Laboratory for Advanced Process Control of Light Industry of the Ministry of Education, School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China 2 Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway
Correspondence should be addressed to Fei Liu; [email protected]
Received 10 July 2013; Accepted 5 September 2013
Academic Editor: Lixian Zhang
Copyright © 2013 Fei Chen 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.
This paper deals with the finite-time stabilization problem for discrete-time Markov jump nonlinear systems with time delays and norm-bounded exogenous disturbance. The nonlinearities in different jump modes are parameterized by neural networks. Subsequently, a linear difference inclusion state space representation for a class of neural networks is established. Based on this, sufficient conditions are derived in terms of linear matrix inequalities to guarantee stochastic finite-time boundedness and stochastic finite-time stabilization of the closed-loop system. A numerical example is illustrated to verify the efficiency of the proposed technique.
1. Introduction
Markov jump systems (MJSs) are an important class of stochastic dynamic systems, which are popular when modeling an abrupt change in the system structure and parameters, such as component failures or repairs, changing subsystem interconnections and environmental disturbance. This family of systems has great practical potential in a variety of fields, such as solar thermal central receivers systems, economic systems, communication systems, manufacturing systems, and networked control systems [1–4]. MJSs have been extensively studied since the pioneering work on quadratic control of MJSs [5], and many achievements have been made on Lyapunov stochastic stability and stabilization in the last three decades [6–18].
However, it is worth noting that the Lyapunov stochastically stable systems may not possess good or expected transient characteristics over a finite-time horizon. In many practical problems, it is of interest to investigate the stability of a system over a finite interval of time. For example, referring to aircraft control, it requests that, during the
execution of a certain task, the state variables should not exceed some threshold under all admissible pilot inputs and in the presence of wind disturbances. Classical control theory does not directly address this requirement, because it focuses mainly on the asymptotic behavior of the system (over an infinite-time interval) and does not usually specify bounds on the trajectories. Therefore, it is necessary to limit the state in an acceptable region and consider finite-time stability (FTS) given by Dorato [19].
The concept of FTS has been further extended into finite-time boundness (FTB) [20, 21], when system possesses bounded exogenous disturbance. A linear matrix inequality (LMI) framework has been established to distinguish FTS and Lyapunov asymptotical stability [22–24]. Compared with Lyapunov stochastically stable condition, FTS relaxes the condition by allowing that the Lyapunov-like function can increase at every sampling time instant. That is why FTS is so attractive and widely used in practical engineering.
As MJSs are considered, a number of results on stochastic FTS or stochastic FTB have been developed [25–28], and recently, the obtained results have been extended to
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Abstract and Applied Analysis
continuous-time MJSs with nonlinearities via fuzzy or neural network approach [24, 29, 30]. In order to make the stochastic systems more manageable and satisfy the requirements for finite-time behavior of a system in engineering fields, it motivates us to investigate the finite-time stability and stabilization problems for a class of MJSs. Furthermore, time delay is a common phenomenon and is inevitable in practice systems [31–33]. Due to the interaction among system dynamics, stochastic jumps, and time delays, the dynamics of MJSs with time delay become more complex than MJSs without time delay and time delay systems without jumps. So far, in comparison with the literatures available for continuous-time nonlinear MJSs with time delays, the corresponding FTS or FTB results for discrete-time nonlinear systems have been relatively few.
It is, therefore, the main purpose of this paper to shorten such a gap by investigating the finite-time stabilization problem for discrete-time nonlinear MJSs with time delays. With neural networks, the nonlinearities of MJSs are approximated firstly by linear difference inclusion under state-space representation. Then, a mode-dependent finite-time controller is developed to make the nonlinear MJSs stochastic finitetime stabilizable for all admissible approximation errors of the neural networks and the norm-bounded external disturbances. The controller gains could be derived by solving a set of LMIs. An attractive feature of the proposed scheme is that the coupling relationship between time delay and given finite-time horizon is explored by obtaining delayindependent conditions.
Notations in this paper are fairly standard. 𝑅𝑛 and 𝑅𝑛×𝑚 denote 𝑛-dimensional Euclidean space and the set of all the 𝑛 × 𝑚 real matrices, respectively; 𝐴𝑇 (or 𝑥𝑇) and 𝐴−1 denote the transpose of the matrix 𝐴 (or the vector 𝑥) and the inverse of the matrix 𝐴, respectively. 𝜆max(𝐴) and 𝜆min(𝐴) denote, respectively, the maximal and minimal eigenvalues of a real matrix 𝐴, ‖𝐴‖ denotes the Euclidean norm of matrix 𝐴, 𝐸{⋅} denotes the mathematics statistical expectation of the stochastic process or vector, 𝑙2[0 𝑁) is the space of summable infinite sequence over [0 𝑁), 𝑃 > 0 stands for a positive-definite matrix, 𝐼 is the unit matrix with appropriate dimensions, and “∗” means the symmetric terms in a symmetric matrix.
2. System Description and Problem Formulation
We consider a nonlinear discrete-time MJS, which can be described by the following mathematical model:
𝑥𝑘+1 = 𝐴 (𝑟𝑘) 𝑥𝑘 + 𝐴𝑑 (𝑟𝑘) 𝑥𝑘−𝑑 + 𝐵 (𝑟𝑘) 𝑢𝑘
+ 𝐵𝑤 (𝑟𝑘) 𝑤𝑘 + 𝐶 (𝑟𝑘) 𝑓 (𝑥𝑘, 𝑟𝑘) ,
(1)
𝑥𝑓 = 𝜑𝑓, 𝑓 ∈ {−𝑑, . . . , 0} , 𝑟 (0) = 𝑟0,
where 𝑥𝑘 ∈ 𝑅𝑛 is the vector of state variables, 𝑢𝑘 ∈ 𝑅𝑚 is the controlled input, 𝑓(⋅) is a discrete nonlinear mapping with
𝑓(0) = 0 but not assumed to be known a prior, and 𝑤𝑘 ∈ 𝑙2𝑞[0 + ∞) is the exogenous disturbances satisfying
𝑁
‖𝑤‖22 = 𝐸 [∑𝑤𝑘𝑇𝑤𝑘] < 𝛿2.
(2)
𝑘=0
For each possible value of 𝑟𝑘 = 𝑖, we denote
𝐴 (𝑟𝑘) = 𝐴𝑖, 𝐴𝑑 (𝑟𝑘) = 𝐴𝑑𝑖, 𝐵 (𝑟𝑘) = 𝐵𝑖,
𝐵𝑤 (𝑟𝑘) = 𝐵𝑤𝑖,
𝐶 (𝑟𝑘) = 𝐶𝑖,
𝑓 (𝑥𝑘, 𝑟𝑘) = 𝑓𝑖 (𝑥𝑘) , (3)
where 𝑟𝑘 is a discrete-state Markov chain taking values in 𝑀 = {1, 2, . . . , 𝑠} with transition probabilities
Prob {𝑟𝑘+1 = 𝑗 | 𝑟𝑘 = 𝑖} = 𝜋𝑖𝑗,
(4)
where 𝜋𝑖𝑗 is the transition probabilities from mode 𝑖 to mode 𝑗 that satisfies
𝑚
𝜋𝑖𝑗 ≥ 0, ∑𝜋𝑖𝑗 = 1, ∀𝑖, 𝑗 ∈ 𝑀.
(5)
𝑗=1
For each mode 𝑖, nonlinear function 𝑓𝑖(𝑥𝑘) is to be parameterized by neural networks. Such parameterization makes sense because any nonlinear function can be approximated arbitrarily well on a compact interval by a neural network. Let the 𝐿-layered perceptrons 𝑁𝑖(𝑥𝑘, 𝑊𝑟1, 𝑊𝑟2, . . . , 𝑊𝑟𝐿) be suitably trained to approximate the nonlinear term 𝑓𝑖(𝑥𝑘), which is described in matrix-vector notation as
𝑁𝑖 (𝑥𝑘, 𝑊𝑖1, 𝑊𝑖2, . . . , 𝑊𝑖𝐿) (6)
= 𝜓𝑖𝐿 [𝑊𝑖𝐿 ⋅ ⋅ ⋅ 𝜓𝑖2 [𝑊𝑖2 [𝜓𝑖1 [𝑊𝑖1𝑥 (𝑡)]]]] ,
where all the weight matrices 𝑊𝑖𝑟 ∈ 𝑅 , 𝑛𝑖𝑟×𝑛𝑖(𝑟−1) 𝑟 = 1, . . ., 𝐿, from the (𝑟 − 1)th layer to the (𝑟 − 𝐿)th layer will be determined via back propagation (BP) procedure [24]; the activation function vector of 𝑟th layer is defined as 𝜓𝑖𝑟[𝜁] = [𝜙𝑖1(𝜍𝑖1), 𝜙𝑖2(𝜍𝑖2), . . . , 𝜙𝑖𝑛𝑟 (𝜍𝑖𝑛𝑟 )]𝑇, where 𝑛𝑟 indicates the neurons of 𝑟th layer and let
1 − 𝑒−𝜍𝑖𝑙/𝑞𝑖𝑙 𝜙𝑖𝑙 (𝜍𝑖𝑙) = 𝜆𝑖𝑙 ( 1 + 𝑒−𝜍𝑖𝑙/𝑞𝑖𝑙 ) ,
𝑞𝑖𝑙, 𝜆𝑖𝑙 > 0, 𝑙 = 1, 2, . . . , 𝑛𝑟. (7)
The maximum and minimum derivatives of activation function 𝜙𝑖𝑙 are defined as follows:
{{{{min 𝜕𝜙𝜕𝑖𝑙𝜁(𝜁𝑖𝑙) , 𝑚 = 0,
𝑠𝑖𝑙 (𝑚, 𝜙𝑖𝑙) = {{ 𝜁𝑖𝑙 {
𝑖𝑙
𝜕𝜙 (𝜁 )
(8)
{max 𝑖𝑙 𝑖𝑙 , 𝑚 = 1.
{ 𝜁𝑖𝑙 𝜕𝜁𝑖𝑙
For 𝑟th layer of neural network, activation function 𝜙𝑖𝑙 can be rewritten as the following min-max form:
𝜙𝑖𝑙 = ℎ𝑖𝑙 (0) 𝑠𝑖𝑙 (0, 𝜙𝑖𝑙) + ℎ𝑖𝑙 (1) 𝑠𝑖𝑙 (1, 𝜙𝑖𝑙) ,
(9)
Abstract and Applied Analysis
3
where ℎ𝑖𝑙(𝑚), 𝑚 = 0, 1, are a set of positive real numbers associated with 𝜙𝑖𝑙 satisfying ℎ𝑖𝑙(𝑚) > 0 and ℎ𝑖𝑙(0)+ℎ𝑖𝑙(1) = 1.
According to the approximation theorem, for given accuracy 𝜌𝑖 > 0, there exist ideal constant weight matrices 𝑊𝑖∗𝑟 defined as
(𝑊𝑖∗1, 𝑊𝑖∗2, . . . , 𝑊𝑖∗𝐿)
= arg min {max 𝑓𝑖 (𝑥𝑘) − 𝑁𝑖 (𝑥𝑘, 𝑊𝑖1, 𝑊𝑖2, . . . , 𝑊𝑖𝐿)} , ( ) 𝑊𝑖∗1 ,𝑊𝑖∗2 ,...,𝑊𝑖∗𝐿 𝑥𝑘∈𝐷 (10)
where 𝐷 is a compact set 𝐷 ∈ 𝑅𝑚, such that max 𝑓𝑖 (𝑥𝑘) − 𝑁𝑖 (𝑥𝑘, 𝑊𝑖∗1, 𝑊𝑖∗2, . . . , 𝑊𝑖∗𝐿) ≤ 𝜌𝑖 𝑥𝑘 .
𝑥𝑘 ∈𝐷
(11)
For each mode 𝑖, denote a set of 𝑛𝑟 dimensional index vectors of the 𝑟th layer as
𝛾𝑛𝑟 = 𝛾𝑛𝑟 (𝜎𝑖) = {𝜎𝑖 ∈ 𝑅𝑛𝑟 | 𝜎𝑖𝑙 ∈ {0, 1} , 𝑙 = 1, . . . , 𝑛𝑟} , (12)
where 𝜎𝑖 is used as a binary indicator. Obviously, the 𝑟th layer with 𝑛𝑟 neurons has 2𝑛𝑟 combinations of binary indicator with 𝑚 = 0, 1, and the elements of index vectors for all 𝐿 layers neural network have 2𝑛𝐿 × ⋅ ⋅ ⋅ × 2𝑛2 × 2𝑛1 combinations in the set
Θ = 𝛾𝑛𝐿 ⊕ ⋅ ⋅ ⋅ ⊕ 𝛾𝑛2 ⊕ 𝛾𝑛1 .
(13)
By using (8) and adopting the compact representation [34], the multilayer neural network (6) can be expressed as follows:
1
[[
[[ [[ ∑ ℎ𝑖11 (𝑘) 𝑠𝑖11 (𝑚, 𝜙𝑖11) × (𝑊𝑖∗1𝑥)𝑖1 ]]]] ]]
𝑁 (𝑥 , 𝑊∗, 𝑊∗, . . . , 𝑊∗) = 𝜓 [[[𝑊∗ ⋅ ⋅ ⋅ 𝜓 [[[𝑊∗ [[[ 𝑚=0
..
]]]]]] ⋅ ⋅ ⋅ ]]]
𝑥 𝑖1 𝑖2
𝑖𝐿
𝑖𝐿 [[ 𝑖𝐿 𝑖2 [[ 𝑖2 [[ 1
.
]]]] ]]
[
[ [ ∑ ℎ (𝑘) 𝑠 (𝑚, 𝜙 ) × (𝑊∗𝑥) ]] ]
(14)
[ [ [𝑚=0 𝑖1𝑛1 𝑖1𝑛1 𝑖1𝑛1 𝑖1 𝑖𝑛1 ]] ]
= ∑ 𝜇𝜎 𝐴𝜎 (𝜎𝑖, 𝜓𝑖, 𝑊𝑖∗) 𝑥𝑘,
𝑖
𝑖
𝜎𝑖 ∈Θ
where
𝐴𝜎 = diag [𝑠𝐿𝑖𝑙 (𝜎𝐿𝑖𝑙, 𝜙𝐿𝑖𝑙)] 𝑊𝐿∗ ⋅ ⋅ ⋅ diag [s2𝑖𝑙 (𝜎2𝑖𝑙, 𝜙2𝑖𝑙)] 𝑖 × 𝑊2∗ diag [𝑠1𝑖𝑙 (𝜎1𝑖𝑙, 𝜙1𝑖𝑙)] 𝑊1∗,
1
1
∑ 𝜇𝜎𝑖 = ∑ ⋅ ⋅ ⋅ ∑ ℎ𝑖𝐿𝑛𝐿 (𝑚𝑖𝐿𝑛𝐿 ) ⋅ ⋅ ⋅ ℎ𝑖𝐿1 (𝑚𝑖𝐿1)
𝜎𝑖 ∈Θ
𝑚𝑖𝐿𝑛𝐿 =0 𝑚𝑖1𝑛1 =0
...
...
𝑚𝑖𝐿1 =0
𝑚𝑖11 =0
⋅ ⋅ ⋅ ℎ𝑖1𝑛1 (𝑚𝑖1𝑛1 ) ⋅ ⋅ ⋅ ℎ𝑖11 (𝑚𝑟11) = 1. (15)
Thus by means of multilayer neural network, the nonlinear MJS (1) is translated into a group of LDIs with error bounds, in which the different inclusion is powered by stochastic Markov process; that is,
𝑥𝑘+1 = [ ∑ 𝜇𝜎 𝐴𝜎 + 𝐴𝑖] 𝑥𝑘 + 𝐴𝑑𝑖𝑥𝑘−𝑑
𝑖
𝑖
[𝜎𝑖∈Θ ] (16)
+ 𝐵𝑖𝑢𝑘 + 𝐵𝑤𝑖𝑤𝑘 + 𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) ,
𝑥𝑓 = 𝜑𝑓, 𝑓 ∈ {−𝑑, . . . , 0} , 𝑟 (0) = 𝑟0,
where Δ𝑓𝑖 (𝑥𝑘) = max 𝑓𝑖 (𝑥𝑘) − 𝑁𝑖 (𝑥𝑘, 𝑊𝑖∗1, 𝑊𝑖∗2, . . . , 𝑊𝑖∗𝐿)
𝑥𝑘 ∈𝐷
≤ 𝜌𝑖 𝑥𝑘 (17)
denotes the approximation errors of networks.
Remark 1. The detailed structure and quantitative size of error dynamics Δ𝑓𝑖(𝑥𝑘) are not needed, but only normbounded assumption is required. This condition is easily satisfied in practical cases, such as bioinformatics system, medical diagnosis, fault diagnosis, and image and pattern recognition. Actually, the approximation error between the target function and the closest neural network function of a given network family can be made as small as desired by increasing the number of nodes [35]. Also the bounds of norm may vary according to different nonlinearities in different modes.
3. Main Results
Based on the LDI model (16) of networks, we consider the following discrete-time state feedback control law for nonlinear stochastic MJS (1):
𝑢𝑘 = 𝐾𝑖𝑥𝑘.
(18)
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Abstract and Applied Analysis
The resulting closed-loop system can be obtained as follows:
𝑥𝑘+1 = 𝐴𝑖𝑥𝑘 + 𝐴𝑑 (𝑟𝑘) 𝑥𝑘−𝑑 + 𝐵𝑤 (𝑟𝑘) 𝑤𝑘 + Δ𝑓𝑖 (𝑥𝑘) ,
𝑥𝑓 = 𝜑𝑓, 𝑓 ∈ {−𝑑, . . . , 0} , 𝑟 (0) = 𝑟0, (19)
where
𝐴𝑖 = ∑ 𝜇𝜎𝑖 𝐴𝜎𝑖 + 𝐴𝑖 + 𝐵𝑖𝐾𝑖.
(20)
𝜎𝑖 ∈Θ
The aim of this paper is to find some sufficient conditions which guarantee stochastic finite-time boundness and stochastic finite-time stabilization of the closed-loop system (19). The general idea of finite-time control can be formalized through the following definitions over a finite-time interval for some given initial conditions.
Definition 2 (stochastic finite-time stability). A discrete-time nonlinear MJS (1) (setting 𝑢𝑘 = 0 and 𝑤𝑘 = 0) is said to be, stochastic finite-time stability (FTS) with respect to given (𝑐1, 𝑐2, 𝐺, 𝑁), where 𝑐2 > 𝑐1 and 𝐺 > 0, if 𝐸{𝑥𝑘𝑇𝐺𝑥𝑘} < 𝑐22, 𝑘 ∈ {1, 2, . . . , 𝑁} whenever max𝑘0−𝑑≤𝑘≤𝑘0 𝐸{𝑥0𝑇𝐺𝑥0} ≤ 𝑐12.
Definition 3 (stochastic finite-time boundness). A discretetime nonlinear MJS (1) (setting 𝑢𝑘 = 0) is said to be of stochastic finite-time boundness (FTB) with respect to (𝑐1, 𝑐2, 𝐺, 𝑁, 𝛿) with 𝑐2 > 𝑐1 and 𝐺 > 0, if 𝐸{𝑥𝑘𝑇𝐺𝑥𝑘} < 𝑐22, 𝑘 ∈ {1, 2, . . . , 𝑁}, whenever max𝑘0−𝑑≤𝑘≤𝑘0 𝐸{𝑥0𝑇𝐺𝑥0} ≤ 𝑐12.
Before proceeding further, we introduce the following lemmas which will be needed for the derivation of our main results.
Lemma 4. The closed-loop system (19) is stochastic FTB with respect to the given (𝑐1, 𝑐2, 𝐺, 𝑁, 𝛿) and scalar 𝛼 ≥ 0, if there exist mode-dependent symmetric positive-definite matrix 𝑃𝑖 and symmetric positive-definite matrices 𝑄 and 𝑆 such that
𝐴𝑇𝑖 𝑃𝑗𝐴𝑖 − (1 + 𝛼) 𝑃𝑖 + 𝑄
[[
∗
[[
[[
∗
[
∗
< 0,
𝐴𝑇𝑖 𝑃𝑗𝐴𝑑𝑖 𝐴𝑇𝑑𝑖𝑃𝑗𝐴𝑑𝑖 − 𝑄
∗ ∗
𝐴𝑇𝑖 𝑃𝑗𝐵𝑤𝑖 𝐴𝑇𝑑𝑖 𝑃𝑗 𝐵𝑤𝑖 𝐵𝑤𝑇𝑖𝑃𝑗𝐵𝑤𝑖 − (1 + 𝛼) 𝑆
∗
𝐴𝑇𝑖 𝑃𝑗𝐶𝑖
]] 𝐴𝑇𝑑𝑖 𝑃𝑗 𝐶𝑖 ]]]] 𝐵𝑤𝑇 𝑖 𝑃𝑗 𝐶𝑖 𝐶𝑖𝑇𝑃𝑗𝐶𝑖 − (1 + 𝛼) 𝑅]
(21)
𝑐12max {𝜆max (𝑃̃𝑖)} + 𝑐12𝑑𝜆max (𝑄̃) + 𝛿2𝜆max (𝑆)
𝑖∈𝑀
+ 𝑐2𝜌2𝜆
(𝑅̃)
<
𝑐22min𝑖∈𝑀 {𝜆min (𝑃̃𝑖)} ,
(22)
2 𝑖 max
(1 + 𝛼)𝑁
where 𝑃̃𝑖 = 𝐺−1/2𝑃𝑖𝐺−1/2, 𝑄̃ = 𝐺−1/2𝑄𝐺−1/2, 𝑅̃ = 𝐺−1/2𝑅𝐺−1/2, and 𝜆max(⋅), 𝜆min(⋅) indicate the maximal and minimal eigenvalues of the augment, respectively.
Proof. For the closed-loop system (19), choose a stochastic Lyapunov function candidate as
𝑘−1
𝑉𝑖 (𝑘) = 𝑥𝑘𝑇𝑃𝑖𝑥𝑘 + ∑ 𝑥𝑓𝑇𝑄𝑥𝑓.
(23)
𝑓=𝑘−𝑑
Simple calculation shows that
𝐸 {𝑉𝑖 (𝑘 + 1)} − 𝑉𝑖 (𝑘) = 𝑥𝑘𝑇 (𝐴𝑇𝑖 𝑃𝑗𝐴𝑖 − 𝑃𝑖 + 𝑄) 𝑥𝑘 + 2𝑥𝑘𝑇𝐴𝑇𝑖 𝑃𝑗𝐴𝑑𝑖𝑥𝑘−𝑑
+ 2𝑥𝑘𝑇𝐴𝑇𝑖 𝑃𝑗𝐵𝑤𝑖𝑤𝑘 + 2𝑥𝑘𝑇𝐴𝑇𝑖 𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) + 𝑥𝑘𝑇−𝑑 (𝐴𝑇𝑑𝑖𝑃𝑗𝐴 𝑑𝑖 − 𝑄) 𝑥𝑘−𝑑 + 2𝑥𝑘𝑇−𝑑𝐴𝑇𝑑𝑖𝑃𝑗𝐵𝑤𝑖𝑤𝑘 + 2𝑥𝑘𝑇−𝑑𝐴𝑇𝑑𝑖𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) + 𝑤𝑘Τ𝐵𝑤𝑇𝑖𝑃𝑗𝐵𝑤𝑖𝑤𝑘 + 2𝑤𝑘𝑇𝐵𝑤𝑇𝑖𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) + Δ𝑓𝑖𝑇 (𝑥𝑘) 𝐶𝑖𝑇𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) = 𝜁𝑘𝑇Ω𝑖𝜁𝑘,
(24)
where
𝑠
𝑃𝑗 ≜ ∑𝜋𝑖𝑗𝑃𝑗,
𝑗=1
𝜁𝑘 = [𝑥𝑘𝑇 𝑥𝑘𝑇−𝑑 𝑤𝑘𝑇 Δ𝑓𝑖𝑇 (𝑥𝑘)]𝑇,
𝐴𝑇𝑖 𝑃𝑗𝐴𝑖 − 𝑃𝑖 + 𝑄
Ω𝑖 = [[[
𝐴𝑇𝑑𝑖 𝑃𝑗 𝐴𝑖 𝐵𝑤𝑇 𝑖 𝑃𝑗 𝐴𝑖
[ 𝐶𝑖𝑇𝑃𝑗𝐴𝑖
∗ 𝐴𝑇𝑑𝑖𝑃𝑗𝐴 𝑑𝑖 − 𝑄
𝐵𝑤𝑇𝑖𝑃𝑗𝐴 𝑑𝑖 𝐶𝑖𝑇𝑃𝑗𝐴 𝑑𝑖
∗
∗ 𝐵𝑤𝑇 𝑖 𝑃𝑗 𝐵𝑤𝑖 𝐶𝑖𝑇 𝑃𝑗 𝐵𝑤𝑖
∗ ∗∗ ]]] .
𝐶𝑖𝑇𝑃𝑗𝐶𝑖] (25)
Conditions (21) and (24) imply that
𝐸 {𝑉𝑖 (𝑘 + 1)}
≤ (1 + 𝛼) 𝑥𝑘𝑇𝑃𝑖𝑥𝑘 + (1 + 𝛼) 𝑤𝑘𝑇𝑆𝑤𝑘
𝑘−1
+ (1 + 𝛼) Δ𝑓𝑖𝑇 (𝑥𝑘) 𝑅Δ𝑓𝑖 (𝑥𝑘) + (1 + 𝛼) ∑ 𝑥𝑓𝑇𝑄𝑥𝑓
𝑓=𝑘−𝑑
= (1 + 𝛼) 𝑉𝑖 (𝑘) + (1 + 𝛼) 𝑤𝑘𝑇𝑆𝑤𝑘
+ (1 + 𝛼) Δ𝑓𝑖𝑇 (𝑥𝑘) 𝑅Δ𝑓𝑖 (𝑥𝑘) . (26)
Noting that 𝛼 ≥ 0, we can obtain from (26) that
𝑘
𝑉𝑖 (𝑘) ≤ (1 + 𝛼)𝑘𝑉𝑖 (0) + ∑ (1 + 𝛼)𝑘−𝑓+1𝑤𝑓𝑇−1𝑆𝑤𝑓−1
𝑓=1
𝑘
+ ∑ (1 + 𝛼)𝑘−𝑓+1𝑐22𝜌𝑖2𝜆max (𝑅̃)
𝑓=1
Abstract and Applied Analysis
−1
= (1 + 𝛼)𝑘 [𝑥0𝑇𝑃𝑖𝑥0 + ∑ 𝑥𝑓𝑇𝑄𝑥𝑓
[
𝑓=−𝑑
𝑘
+ ∑ (1 + 𝛼)1−𝑓𝑤𝑓𝑇−1𝑆𝑤𝑓−1
𝑓=1
𝑘
+ ∑ (1 + 𝛼)1−𝑓𝑐22𝜌𝑖2𝜆max (𝑅̃)]
𝑓=1
]
≤ (1 + 𝛼)𝑁 [𝑐12max {𝜆max (𝑃̃𝑖)} + 𝑐12𝑑𝜆max (𝑄̃)
𝑖∈𝑀
+ 𝛿2𝜆max (𝑆) + 𝑐22𝜌𝑖2𝜆max (𝑅̃)] .
(27)
Note that
𝑘−1
𝑉𝑖 (𝑘) = 𝑥𝑘𝑇𝑃𝑖𝑥𝑘 + ∑ 𝑥𝑓𝑇𝑄𝑥𝑓
𝑓=𝑘−𝑑
≥ 𝑥𝑇𝑃 𝑥
(28)
𝑘 𝑖𝑘
≥ min {𝜆min (𝑃̃𝑖)} 𝑥𝑘𝑇𝐺𝑥𝑘.
𝑖∈𝑀
According to (27)-(28), one has
𝑥𝑘𝑇𝐺𝑥𝑘 ≤ ((1 + 𝛼)𝑁 (𝑐12max {𝜆max (𝑃̃𝑖)} + 𝑐12𝑑𝜆max (𝑄̃)
𝑖∈𝑀
+𝛿2𝜆max (𝑆) + 𝑐22𝜌𝑖2𝜆max (𝑅̃) ) )
−1
× (min {𝜆min (𝑃̃𝑖)}) .
𝑖∈𝑀
(29)
Condition (19) implies that, for 𝑘 ∈ {1, 2, . . . , 𝑁}, 𝐸{𝑥𝑘𝑇𝐺𝑥𝑘} < 𝑐22. This completes the proof.
Now, we direct our attention to present a solution to the problem of finite-time stabilizing controller design. Such controller is provided by the following theorem.
Theorem 5. The closed-loop system (19) is stochastic finitetime stabilizable via state feedback with respect to the given (𝑐1, 𝑐2, 𝐺, 𝑁, 𝛿) and scalar 𝛼 ≥ 0, if there exist matrices 𝑋𝑖 = 𝑋𝑖𝑇 > 0, 𝑌𝑖, 𝐻 = 𝐻𝑇 > 0, 𝑆 = 𝑆𝑇 > 0, and 𝑅 = 𝑅𝑇 > 0 and scalars 𝜆1, 𝜆2, 𝜆3, 𝜆4 > 0 such that
− (1 + 𝛼) 𝑋𝑖 [[ 𝑁1𝑖 [[[ 00 [ 𝑋𝑖
< 0,
𝑁1𝑇𝑖 −𝑀5𝑖 + 𝑁5𝑖
𝑀3𝑇𝑖 𝑀4𝑇𝑖
0
0 𝑀3𝑖 − (1 + 𝛼) 𝑆
0 0
0 𝑀4𝑖
0
− (1 + 𝛼) 𝑅 0
𝑋𝑖 0 ]] 00 ]]] −𝐻]
(30)
5
𝜆1𝐺−1 < 𝑋𝑖 < 𝐺−1,
𝜆2𝐺−1 < 𝐻,
𝑆 < 𝜆3𝐼,
𝑅 < 𝜆4𝐺, (31)
[− 𝑐22 + 𝛿2𝜆3 + 𝑐22𝜌𝑖2𝜆4 𝑐1 √𝑑𝑐1]
[ (1 + 𝛼)𝑁
[
𝑐
−𝜆 0 ]] < 0. (32)
1
1
[
√𝑑𝑐1
0 −𝜆2 ]
Proof. By using Schur complement, from condition (21) in Lemma 4, it follows that
− (1 + 𝛼) 𝑃𝑖 + 𝑄 ∗
∗
∗
[[
0
−𝑄 ∗
∗
[[[ 00
0 − (1 + 𝛼) 𝑆 ∗
0
0 − (1 + 𝛼) 𝑅
∗ ∗ ]] ∗∗ ]]] ≤ 0,
[
𝑀1𝑖
𝑀2𝑖 𝑀3𝑖
𝑀4𝑖 −𝑀5𝑖] (33)
where
𝑇
𝑇𝑇
𝑀1𝑖 = [√𝜋𝑖1𝐴𝑖 , . . . , √𝜋𝑖𝑠𝐴𝑖 ] ,
𝑀2𝑖 = [√𝜋𝑖1𝐴𝑇𝑑𝑖, . . . , √𝜋𝑖𝑠𝐴𝑇𝑑𝑖]𝑇,
𝑀3𝑖 = [√𝜋𝑖1𝐵𝑤𝑇𝑖, . . . , √𝜋𝑖𝑠𝐵𝑤𝑇𝑖]𝑇,
(34)
𝑀4𝑖 = [√𝜋𝑖1𝐶𝑖𝑇, . . . , √𝜋𝑖𝑠𝐶𝑖𝑇]𝑇,
𝑀5𝑖 = diag {𝑃1−1, . . . , 𝑃𝑠−1} .
Performing matrix elementary transformation to the above inequality, we have
− (1 + 𝛼) 𝑃𝑖 + 𝑄
[[
𝑀1𝑖
[[[ 00
[
0
𝑀1𝑇𝑖 −𝑀5𝑖 𝑀3𝑇𝑖 𝑀4𝑇𝑖 𝑀2𝑇𝑖
0 𝑀3𝑖 − (1 + 𝛼) 𝑆
0
0
0 𝑀4𝑖
0
− (1 + 𝛼) 𝑅
0
0 𝑀2𝑖]]
00 ]]] ≤ 0.
−𝑄 ] (35)
Performing a congruence to the above condition by diag {𝑃𝑖−1 𝐼 𝐼 𝐼 𝐼}, using Schur complement, and letting 𝑋𝑖 = 𝑃𝑖−1 and 𝑌𝑖 = 𝐾𝑖𝑋𝑖, we get
− (1 + 𝛼) 𝑋𝑖 + 𝑋𝑖𝑄𝑋𝑖 [[ 𝑁01𝑖
[
0
𝑁1𝑇𝑖 −𝑀5𝑖 + 𝑁5𝑖
𝑀3𝑇𝑖 𝑀4𝑇𝑖
0 𝑀3𝑖 − (1 + 𝛼) 𝑆
0
0 𝑀04𝑖 ]]
− (1 + 𝛼) 𝑅]
≤ 0, (36)
6
Abstract and Applied Analysis
where
𝑁1𝑖
=
[√𝜋𝑖1(𝐴̃𝑖𝑋𝑖
+
𝑇
𝐵𝑖𝑌𝑖) ,
.
.
.
,
√𝜋𝑖𝑠(𝐴̃𝑖𝑋𝑖
+
𝑇𝑇
𝐵𝑖𝑌𝑖) ] ,
𝑁5𝑖
𝜋𝑖1 𝐴 𝑑𝑖 𝐻𝐴𝑇𝑑𝑖
√𝜋𝑖1√𝜋𝑖2𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 ⋅ ⋅ ⋅ √𝜋𝑖1√𝜋𝑖𝑠𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖
[[[√𝜋𝑖2√𝜋𝑖1𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 𝜋𝑖2𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 ⋅ ⋅ ⋅ √𝜋𝑖2√𝜋𝑖𝑠𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖]]]
= [[
]] ,
[ ...
... d ... ]
[√𝜋𝑖𝑠√𝜋𝑖1𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 √𝜋𝑖𝑠√𝜋𝑖2𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖
𝜋𝑖𝑠𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 ]
𝐴̃𝑖 = ∑ 𝜇𝜎𝑖 𝐴𝜎𝑖 + 𝐴𝑖.
𝜎𝑖 ∈Θ
(37)
By using Schur complement to (36) and letting 𝐻 = 𝑄−1, we
obtain the linear matrix inequality (30) in Theorem 5.
On the other hand, we consider
𝜆 (𝑋̃ ) = 1 ,
max 𝑖 𝜆min (𝑃̃𝑖)
(38)
𝑋̃𝑖 = 𝑃̃𝑖−1 = 𝐺1/2𝑋𝑖𝐺1/2.
Condition (22) follows that
𝑐12
+ 𝑐2𝑑𝜆 (𝑄̃) + 𝛿2𝜆 (S)
min𝑖∈𝑀 {𝜆min (𝑋̃𝑖)} 1 max
max
(39)
<
𝑐22
.
max𝑖∈𝑀 {𝜆max (𝑋̃𝑖)} (1 + 𝛼)𝑁
It is easy to check that the above inequality is guaranteed by imposing the following conditions
𝜆max (𝑋̃𝑖) < 1, 𝜆1 < 𝜆min (𝑋̃𝑖) ,
𝜆max (𝑄̃) < 𝜆2, 𝜆max (𝑆) < 𝜆3,
(40)
𝑐12 + 𝑐2𝑑𝜆 + 𝛿2𝜆 < 𝑐22
𝜆1 1 2
3 (1 + 𝛼)𝑁
which are equivalent to conditions (31)-(32). This completes the proof.
Remark 6. It is worth pointing out that Theorem 5 is not a delay-dependent sufficient criterion, which is conservative when the delay is small. Delay-dependent result can be developed in the same way by choosing a Lyapunov functional that includes more entries, as was done in [36], or delay fractioning approach that can be employed as was done in [31–33].
Remark 7. The coupling relationship between time delay and given finite-time horizon of the underlying system is obtained through a finite-time stable constraint (32) in Theorem 5. From condition (32), it can be seen that, in given finite-time horizon, if the time delay 𝑑 is larger, constraint (32) is more difficult to be satisfied, which means that the existence of time delay increases the instability of system.
4. Numerical Example
Consider discrete-time Markov jump nonlinear system (1) with three operation modes and the following data:
𝐴1 = [00..8480 −−00..0752] , 𝐴𝑑1 = [−00.2.2 00.1.15] ,
𝐵1 = [21] , 𝐵𝑤1 = [00..45] , 𝐶1 = [00.1] ,
𝐴2 = [0.280 00..2342] ,
𝐴
𝑑2
=
[
−0.6 0.2
00..46] ,
𝐵2 = [−11] , 𝐵𝑤2 = [00..26] , 𝐶2 = [00.3] , (41)
𝐴3 = [−0.08.08 00..1646] , 𝐴𝑑3 = [−00.2.3 00..15] ,
𝐵3 = [11] , 𝐵𝑤2 = [00..13] , 𝐶3 = [00.5] ,
𝑓1 (𝑥𝑘) = 𝑓2 (𝑥𝑘) = 𝑓3 (𝑥𝑘) = sin (𝑥1𝑘) cos (𝑥2𝑘) .
Now, a single hidden layer neural network with 2 hidden
neurons was chosen to approximate the nonlinear functions 𝑓𝑖(𝑥𝑘). All parameters of activation functions associated with the hidden layer were chosen to be 𝑞𝑖𝑙 = 0.5 and 𝜆𝑖𝑙 = 1. For these activation functions, we have 𝑠𝑖𝑙(0, 𝜙𝑖𝑙) = 0 and 𝑠𝑖𝑙(1, 𝜙𝑖𝑙) = 1. The connection weights are trained offline by using BP algorithm. The initial weights and state vector are placed by uniformly distributed random numbers in [−1 1].
After 1000 training steps, the optimal approximation weights
are as follows:
𝑊∗ = [−0.86017 −0.81881] ,
1 −0.95025 0.96405
(42)
𝑊2∗ = [−0.57752 −0.58342] .
The upper bound of approximation error is estimated as 𝜌𝑖 = 0.022. Obviously, in this case, we have Θ = 22 × 21. According to (15), 𝐴𝜎𝑖 can be obtained as follows:
𝐴 𝑖1 = 𝐴 𝑖2 = 𝐴 𝑖3 = 𝐴 𝑖4 = 𝐴 𝑖5 = 𝐴 1⊕[0,0,0]𝑇 = 𝐴0⊕[𝑖,𝑗,𝑘]𝑇 [00 00] , (𝑖, 𝑗, 𝑘 ∈ {0, 1}) ,
𝐴𝑖6 = 𝐴1⊕[1,0]𝑇 = [0.490677 0.470288] ,
(43)
𝐴𝑖7 = 𝐴1⊕[0,1]𝑇 = [0.550439 −0.506245] ,
𝐴𝑖8 = 𝐴1⊕[1,1]𝑇 = [1.00512 −0.0809567] .
The initial state and initial mode are taken as 𝑥0 = [−0.3 0.4]𝑇 and 𝑟0 = 1, respectively. The iterative step is taken as 𝑁 = 7. The mode path from time step 0 to time
Abstract and Applied Analysis
m
3.5
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
Time step
Figure 1: Jump modes.
x2
7
3
2
1
0
−1
−2
−3
−3
−2
−1
0
1
2
3
x1
Figure 3: State trajectory under finite-time control.
x2
15
10
5
0
−5
−10
−15
−5
0
5
10
15
20
25
x1
Figure 2: State trajectory of free MJS.
step 7 is generated randomly and it is shown in Figure 1. Let 𝑐1 = 0.5, 𝑐2 = 2, 𝑁 = 7, 𝐺 = I, 𝑑 = 0.5, 𝛼 = 0.5, and 𝛿2 = 1. By solving the matrix inequalities in Theorem 5, we have the
following controller gains:
𝐾1 = [−0.9304 −0.0683] ,
𝐾2 = [−1.7231 0.3654] ,
(44)
𝐾3 = [1.1486 −0.1588] .
The state trajectories of the free and controlled MJLS (16) are drawn in Figures 2 and 3, respectively. It could be seen that the free MJLS (16) is not stochastic FTB because the trajectory exceeds the given bound 𝑐22. However, the trajectory is limited between the two ellipsoids regions by employing the proposed control move which satisfactorily justify that the closed-loop MJLS (16) is stochastic FTB.
It should be pointed out that, in the simulation example, as long as the choice of initial condition is satisfied with ‖𝑥0𝑇𝑅𝑟𝑥0‖ ≤ 𝑐1, then the system is robustly finite-time stabilizable; that is, system trajectories stay within a given bound.
5. Conclusions
The finite-time stabilization problem for discrete-time Markovian jump nonlinear system with time delay and normbounded exogenous disturbance is investigated in this paper. The nonlinearities are parameterized by multilayer neural network and the relationship between time delay and given finite-time horizon is explored with delay-independent conditions. The proposed framework is versatile and can accommodate a number of challenging design problems including finite-time control and filtering of discrete-time or continuous-time nonlinear MJS with parameter uncertainties, time delays, and so on. The future work can consider some delay-dependent approaches or delay fractioning approaches to reduce the conservativeness introduced by time delay.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (Grant no. 61104121) and the 111 Project (Grant no. B12018), and the third author would also like to thank the Alexander-von-Humboldt Foundation for providing the support to this research.
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Abstract and Applied Analysis
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Research Article Neural Network Based Finite-Time Stabilization for Discrete-Time Markov Jump Nonlinear Systems with Time Delays
Fei Chen,1 Fei Liu,1 and Hamid Reza Karimi2
1 Key Laboratory for Advanced Process Control of Light Industry of the Ministry of Education, School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China 2 Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway
Correspondence should be addressed to Fei Liu; [email protected]
Received 10 July 2013; Accepted 5 September 2013
Academic Editor: Lixian Zhang
Copyright © 2013 Fei Chen 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.
This paper deals with the finite-time stabilization problem for discrete-time Markov jump nonlinear systems with time delays and norm-bounded exogenous disturbance. The nonlinearities in different jump modes are parameterized by neural networks. Subsequently, a linear difference inclusion state space representation for a class of neural networks is established. Based on this, sufficient conditions are derived in terms of linear matrix inequalities to guarantee stochastic finite-time boundedness and stochastic finite-time stabilization of the closed-loop system. A numerical example is illustrated to verify the efficiency of the proposed technique.
1. Introduction
Markov jump systems (MJSs) are an important class of stochastic dynamic systems, which are popular when modeling an abrupt change in the system structure and parameters, such as component failures or repairs, changing subsystem interconnections and environmental disturbance. This family of systems has great practical potential in a variety of fields, such as solar thermal central receivers systems, economic systems, communication systems, manufacturing systems, and networked control systems [1–4]. MJSs have been extensively studied since the pioneering work on quadratic control of MJSs [5], and many achievements have been made on Lyapunov stochastic stability and stabilization in the last three decades [6–18].
However, it is worth noting that the Lyapunov stochastically stable systems may not possess good or expected transient characteristics over a finite-time horizon. In many practical problems, it is of interest to investigate the stability of a system over a finite interval of time. For example, referring to aircraft control, it requests that, during the
execution of a certain task, the state variables should not exceed some threshold under all admissible pilot inputs and in the presence of wind disturbances. Classical control theory does not directly address this requirement, because it focuses mainly on the asymptotic behavior of the system (over an infinite-time interval) and does not usually specify bounds on the trajectories. Therefore, it is necessary to limit the state in an acceptable region and consider finite-time stability (FTS) given by Dorato [19].
The concept of FTS has been further extended into finite-time boundness (FTB) [20, 21], when system possesses bounded exogenous disturbance. A linear matrix inequality (LMI) framework has been established to distinguish FTS and Lyapunov asymptotical stability [22–24]. Compared with Lyapunov stochastically stable condition, FTS relaxes the condition by allowing that the Lyapunov-like function can increase at every sampling time instant. That is why FTS is so attractive and widely used in practical engineering.
As MJSs are considered, a number of results on stochastic FTS or stochastic FTB have been developed [25–28], and recently, the obtained results have been extended to
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Abstract and Applied Analysis
continuous-time MJSs with nonlinearities via fuzzy or neural network approach [24, 29, 30]. In order to make the stochastic systems more manageable and satisfy the requirements for finite-time behavior of a system in engineering fields, it motivates us to investigate the finite-time stability and stabilization problems for a class of MJSs. Furthermore, time delay is a common phenomenon and is inevitable in practice systems [31–33]. Due to the interaction among system dynamics, stochastic jumps, and time delays, the dynamics of MJSs with time delay become more complex than MJSs without time delay and time delay systems without jumps. So far, in comparison with the literatures available for continuous-time nonlinear MJSs with time delays, the corresponding FTS or FTB results for discrete-time nonlinear systems have been relatively few.
It is, therefore, the main purpose of this paper to shorten such a gap by investigating the finite-time stabilization problem for discrete-time nonlinear MJSs with time delays. With neural networks, the nonlinearities of MJSs are approximated firstly by linear difference inclusion under state-space representation. Then, a mode-dependent finite-time controller is developed to make the nonlinear MJSs stochastic finitetime stabilizable for all admissible approximation errors of the neural networks and the norm-bounded external disturbances. The controller gains could be derived by solving a set of LMIs. An attractive feature of the proposed scheme is that the coupling relationship between time delay and given finite-time horizon is explored by obtaining delayindependent conditions.
Notations in this paper are fairly standard. 𝑅𝑛 and 𝑅𝑛×𝑚 denote 𝑛-dimensional Euclidean space and the set of all the 𝑛 × 𝑚 real matrices, respectively; 𝐴𝑇 (or 𝑥𝑇) and 𝐴−1 denote the transpose of the matrix 𝐴 (or the vector 𝑥) and the inverse of the matrix 𝐴, respectively. 𝜆max(𝐴) and 𝜆min(𝐴) denote, respectively, the maximal and minimal eigenvalues of a real matrix 𝐴, ‖𝐴‖ denotes the Euclidean norm of matrix 𝐴, 𝐸{⋅} denotes the mathematics statistical expectation of the stochastic process or vector, 𝑙2[0 𝑁) is the space of summable infinite sequence over [0 𝑁), 𝑃 > 0 stands for a positive-definite matrix, 𝐼 is the unit matrix with appropriate dimensions, and “∗” means the symmetric terms in a symmetric matrix.
2. System Description and Problem Formulation
We consider a nonlinear discrete-time MJS, which can be described by the following mathematical model:
𝑥𝑘+1 = 𝐴 (𝑟𝑘) 𝑥𝑘 + 𝐴𝑑 (𝑟𝑘) 𝑥𝑘−𝑑 + 𝐵 (𝑟𝑘) 𝑢𝑘
+ 𝐵𝑤 (𝑟𝑘) 𝑤𝑘 + 𝐶 (𝑟𝑘) 𝑓 (𝑥𝑘, 𝑟𝑘) ,
(1)
𝑥𝑓 = 𝜑𝑓, 𝑓 ∈ {−𝑑, . . . , 0} , 𝑟 (0) = 𝑟0,
where 𝑥𝑘 ∈ 𝑅𝑛 is the vector of state variables, 𝑢𝑘 ∈ 𝑅𝑚 is the controlled input, 𝑓(⋅) is a discrete nonlinear mapping with
𝑓(0) = 0 but not assumed to be known a prior, and 𝑤𝑘 ∈ 𝑙2𝑞[0 + ∞) is the exogenous disturbances satisfying
𝑁
‖𝑤‖22 = 𝐸 [∑𝑤𝑘𝑇𝑤𝑘] < 𝛿2.
(2)
𝑘=0
For each possible value of 𝑟𝑘 = 𝑖, we denote
𝐴 (𝑟𝑘) = 𝐴𝑖, 𝐴𝑑 (𝑟𝑘) = 𝐴𝑑𝑖, 𝐵 (𝑟𝑘) = 𝐵𝑖,
𝐵𝑤 (𝑟𝑘) = 𝐵𝑤𝑖,
𝐶 (𝑟𝑘) = 𝐶𝑖,
𝑓 (𝑥𝑘, 𝑟𝑘) = 𝑓𝑖 (𝑥𝑘) , (3)
where 𝑟𝑘 is a discrete-state Markov chain taking values in 𝑀 = {1, 2, . . . , 𝑠} with transition probabilities
Prob {𝑟𝑘+1 = 𝑗 | 𝑟𝑘 = 𝑖} = 𝜋𝑖𝑗,
(4)
where 𝜋𝑖𝑗 is the transition probabilities from mode 𝑖 to mode 𝑗 that satisfies
𝑚
𝜋𝑖𝑗 ≥ 0, ∑𝜋𝑖𝑗 = 1, ∀𝑖, 𝑗 ∈ 𝑀.
(5)
𝑗=1
For each mode 𝑖, nonlinear function 𝑓𝑖(𝑥𝑘) is to be parameterized by neural networks. Such parameterization makes sense because any nonlinear function can be approximated arbitrarily well on a compact interval by a neural network. Let the 𝐿-layered perceptrons 𝑁𝑖(𝑥𝑘, 𝑊𝑟1, 𝑊𝑟2, . . . , 𝑊𝑟𝐿) be suitably trained to approximate the nonlinear term 𝑓𝑖(𝑥𝑘), which is described in matrix-vector notation as
𝑁𝑖 (𝑥𝑘, 𝑊𝑖1, 𝑊𝑖2, . . . , 𝑊𝑖𝐿) (6)
= 𝜓𝑖𝐿 [𝑊𝑖𝐿 ⋅ ⋅ ⋅ 𝜓𝑖2 [𝑊𝑖2 [𝜓𝑖1 [𝑊𝑖1𝑥 (𝑡)]]]] ,
where all the weight matrices 𝑊𝑖𝑟 ∈ 𝑅 , 𝑛𝑖𝑟×𝑛𝑖(𝑟−1) 𝑟 = 1, . . ., 𝐿, from the (𝑟 − 1)th layer to the (𝑟 − 𝐿)th layer will be determined via back propagation (BP) procedure [24]; the activation function vector of 𝑟th layer is defined as 𝜓𝑖𝑟[𝜁] = [𝜙𝑖1(𝜍𝑖1), 𝜙𝑖2(𝜍𝑖2), . . . , 𝜙𝑖𝑛𝑟 (𝜍𝑖𝑛𝑟 )]𝑇, where 𝑛𝑟 indicates the neurons of 𝑟th layer and let
1 − 𝑒−𝜍𝑖𝑙/𝑞𝑖𝑙 𝜙𝑖𝑙 (𝜍𝑖𝑙) = 𝜆𝑖𝑙 ( 1 + 𝑒−𝜍𝑖𝑙/𝑞𝑖𝑙 ) ,
𝑞𝑖𝑙, 𝜆𝑖𝑙 > 0, 𝑙 = 1, 2, . . . , 𝑛𝑟. (7)
The maximum and minimum derivatives of activation function 𝜙𝑖𝑙 are defined as follows:
{{{{min 𝜕𝜙𝜕𝑖𝑙𝜁(𝜁𝑖𝑙) , 𝑚 = 0,
𝑠𝑖𝑙 (𝑚, 𝜙𝑖𝑙) = {{ 𝜁𝑖𝑙 {
𝑖𝑙
𝜕𝜙 (𝜁 )
(8)
{max 𝑖𝑙 𝑖𝑙 , 𝑚 = 1.
{ 𝜁𝑖𝑙 𝜕𝜁𝑖𝑙
For 𝑟th layer of neural network, activation function 𝜙𝑖𝑙 can be rewritten as the following min-max form:
𝜙𝑖𝑙 = ℎ𝑖𝑙 (0) 𝑠𝑖𝑙 (0, 𝜙𝑖𝑙) + ℎ𝑖𝑙 (1) 𝑠𝑖𝑙 (1, 𝜙𝑖𝑙) ,
(9)
Abstract and Applied Analysis
3
where ℎ𝑖𝑙(𝑚), 𝑚 = 0, 1, are a set of positive real numbers associated with 𝜙𝑖𝑙 satisfying ℎ𝑖𝑙(𝑚) > 0 and ℎ𝑖𝑙(0)+ℎ𝑖𝑙(1) = 1.
According to the approximation theorem, for given accuracy 𝜌𝑖 > 0, there exist ideal constant weight matrices 𝑊𝑖∗𝑟 defined as
(𝑊𝑖∗1, 𝑊𝑖∗2, . . . , 𝑊𝑖∗𝐿)
= arg min {max 𝑓𝑖 (𝑥𝑘) − 𝑁𝑖 (𝑥𝑘, 𝑊𝑖1, 𝑊𝑖2, . . . , 𝑊𝑖𝐿)} , ( ) 𝑊𝑖∗1 ,𝑊𝑖∗2 ,...,𝑊𝑖∗𝐿 𝑥𝑘∈𝐷 (10)
where 𝐷 is a compact set 𝐷 ∈ 𝑅𝑚, such that max 𝑓𝑖 (𝑥𝑘) − 𝑁𝑖 (𝑥𝑘, 𝑊𝑖∗1, 𝑊𝑖∗2, . . . , 𝑊𝑖∗𝐿) ≤ 𝜌𝑖 𝑥𝑘 .
𝑥𝑘 ∈𝐷
(11)
For each mode 𝑖, denote a set of 𝑛𝑟 dimensional index vectors of the 𝑟th layer as
𝛾𝑛𝑟 = 𝛾𝑛𝑟 (𝜎𝑖) = {𝜎𝑖 ∈ 𝑅𝑛𝑟 | 𝜎𝑖𝑙 ∈ {0, 1} , 𝑙 = 1, . . . , 𝑛𝑟} , (12)
where 𝜎𝑖 is used as a binary indicator. Obviously, the 𝑟th layer with 𝑛𝑟 neurons has 2𝑛𝑟 combinations of binary indicator with 𝑚 = 0, 1, and the elements of index vectors for all 𝐿 layers neural network have 2𝑛𝐿 × ⋅ ⋅ ⋅ × 2𝑛2 × 2𝑛1 combinations in the set
Θ = 𝛾𝑛𝐿 ⊕ ⋅ ⋅ ⋅ ⊕ 𝛾𝑛2 ⊕ 𝛾𝑛1 .
(13)
By using (8) and adopting the compact representation [34], the multilayer neural network (6) can be expressed as follows:
1
[[
[[ [[ ∑ ℎ𝑖11 (𝑘) 𝑠𝑖11 (𝑚, 𝜙𝑖11) × (𝑊𝑖∗1𝑥)𝑖1 ]]]] ]]
𝑁 (𝑥 , 𝑊∗, 𝑊∗, . . . , 𝑊∗) = 𝜓 [[[𝑊∗ ⋅ ⋅ ⋅ 𝜓 [[[𝑊∗ [[[ 𝑚=0
..
]]]]]] ⋅ ⋅ ⋅ ]]]
𝑥 𝑖1 𝑖2
𝑖𝐿
𝑖𝐿 [[ 𝑖𝐿 𝑖2 [[ 𝑖2 [[ 1
.
]]]] ]]
[
[ [ ∑ ℎ (𝑘) 𝑠 (𝑚, 𝜙 ) × (𝑊∗𝑥) ]] ]
(14)
[ [ [𝑚=0 𝑖1𝑛1 𝑖1𝑛1 𝑖1𝑛1 𝑖1 𝑖𝑛1 ]] ]
= ∑ 𝜇𝜎 𝐴𝜎 (𝜎𝑖, 𝜓𝑖, 𝑊𝑖∗) 𝑥𝑘,
𝑖
𝑖
𝜎𝑖 ∈Θ
where
𝐴𝜎 = diag [𝑠𝐿𝑖𝑙 (𝜎𝐿𝑖𝑙, 𝜙𝐿𝑖𝑙)] 𝑊𝐿∗ ⋅ ⋅ ⋅ diag [s2𝑖𝑙 (𝜎2𝑖𝑙, 𝜙2𝑖𝑙)] 𝑖 × 𝑊2∗ diag [𝑠1𝑖𝑙 (𝜎1𝑖𝑙, 𝜙1𝑖𝑙)] 𝑊1∗,
1
1
∑ 𝜇𝜎𝑖 = ∑ ⋅ ⋅ ⋅ ∑ ℎ𝑖𝐿𝑛𝐿 (𝑚𝑖𝐿𝑛𝐿 ) ⋅ ⋅ ⋅ ℎ𝑖𝐿1 (𝑚𝑖𝐿1)
𝜎𝑖 ∈Θ
𝑚𝑖𝐿𝑛𝐿 =0 𝑚𝑖1𝑛1 =0
...
...
𝑚𝑖𝐿1 =0
𝑚𝑖11 =0
⋅ ⋅ ⋅ ℎ𝑖1𝑛1 (𝑚𝑖1𝑛1 ) ⋅ ⋅ ⋅ ℎ𝑖11 (𝑚𝑟11) = 1. (15)
Thus by means of multilayer neural network, the nonlinear MJS (1) is translated into a group of LDIs with error bounds, in which the different inclusion is powered by stochastic Markov process; that is,
𝑥𝑘+1 = [ ∑ 𝜇𝜎 𝐴𝜎 + 𝐴𝑖] 𝑥𝑘 + 𝐴𝑑𝑖𝑥𝑘−𝑑
𝑖
𝑖
[𝜎𝑖∈Θ ] (16)
+ 𝐵𝑖𝑢𝑘 + 𝐵𝑤𝑖𝑤𝑘 + 𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) ,
𝑥𝑓 = 𝜑𝑓, 𝑓 ∈ {−𝑑, . . . , 0} , 𝑟 (0) = 𝑟0,
where Δ𝑓𝑖 (𝑥𝑘) = max 𝑓𝑖 (𝑥𝑘) − 𝑁𝑖 (𝑥𝑘, 𝑊𝑖∗1, 𝑊𝑖∗2, . . . , 𝑊𝑖∗𝐿)
𝑥𝑘 ∈𝐷
≤ 𝜌𝑖 𝑥𝑘 (17)
denotes the approximation errors of networks.
Remark 1. The detailed structure and quantitative size of error dynamics Δ𝑓𝑖(𝑥𝑘) are not needed, but only normbounded assumption is required. This condition is easily satisfied in practical cases, such as bioinformatics system, medical diagnosis, fault diagnosis, and image and pattern recognition. Actually, the approximation error between the target function and the closest neural network function of a given network family can be made as small as desired by increasing the number of nodes [35]. Also the bounds of norm may vary according to different nonlinearities in different modes.
3. Main Results
Based on the LDI model (16) of networks, we consider the following discrete-time state feedback control law for nonlinear stochastic MJS (1):
𝑢𝑘 = 𝐾𝑖𝑥𝑘.
(18)
4
Abstract and Applied Analysis
The resulting closed-loop system can be obtained as follows:
𝑥𝑘+1 = 𝐴𝑖𝑥𝑘 + 𝐴𝑑 (𝑟𝑘) 𝑥𝑘−𝑑 + 𝐵𝑤 (𝑟𝑘) 𝑤𝑘 + Δ𝑓𝑖 (𝑥𝑘) ,
𝑥𝑓 = 𝜑𝑓, 𝑓 ∈ {−𝑑, . . . , 0} , 𝑟 (0) = 𝑟0, (19)
where
𝐴𝑖 = ∑ 𝜇𝜎𝑖 𝐴𝜎𝑖 + 𝐴𝑖 + 𝐵𝑖𝐾𝑖.
(20)
𝜎𝑖 ∈Θ
The aim of this paper is to find some sufficient conditions which guarantee stochastic finite-time boundness and stochastic finite-time stabilization of the closed-loop system (19). The general idea of finite-time control can be formalized through the following definitions over a finite-time interval for some given initial conditions.
Definition 2 (stochastic finite-time stability). A discrete-time nonlinear MJS (1) (setting 𝑢𝑘 = 0 and 𝑤𝑘 = 0) is said to be, stochastic finite-time stability (FTS) with respect to given (𝑐1, 𝑐2, 𝐺, 𝑁), where 𝑐2 > 𝑐1 and 𝐺 > 0, if 𝐸{𝑥𝑘𝑇𝐺𝑥𝑘} < 𝑐22, 𝑘 ∈ {1, 2, . . . , 𝑁} whenever max𝑘0−𝑑≤𝑘≤𝑘0 𝐸{𝑥0𝑇𝐺𝑥0} ≤ 𝑐12.
Definition 3 (stochastic finite-time boundness). A discretetime nonlinear MJS (1) (setting 𝑢𝑘 = 0) is said to be of stochastic finite-time boundness (FTB) with respect to (𝑐1, 𝑐2, 𝐺, 𝑁, 𝛿) with 𝑐2 > 𝑐1 and 𝐺 > 0, if 𝐸{𝑥𝑘𝑇𝐺𝑥𝑘} < 𝑐22, 𝑘 ∈ {1, 2, . . . , 𝑁}, whenever max𝑘0−𝑑≤𝑘≤𝑘0 𝐸{𝑥0𝑇𝐺𝑥0} ≤ 𝑐12.
Before proceeding further, we introduce the following lemmas which will be needed for the derivation of our main results.
Lemma 4. The closed-loop system (19) is stochastic FTB with respect to the given (𝑐1, 𝑐2, 𝐺, 𝑁, 𝛿) and scalar 𝛼 ≥ 0, if there exist mode-dependent symmetric positive-definite matrix 𝑃𝑖 and symmetric positive-definite matrices 𝑄 and 𝑆 such that
𝐴𝑇𝑖 𝑃𝑗𝐴𝑖 − (1 + 𝛼) 𝑃𝑖 + 𝑄
[[
∗
[[
[[
∗
[
∗
< 0,
𝐴𝑇𝑖 𝑃𝑗𝐴𝑑𝑖 𝐴𝑇𝑑𝑖𝑃𝑗𝐴𝑑𝑖 − 𝑄
∗ ∗
𝐴𝑇𝑖 𝑃𝑗𝐵𝑤𝑖 𝐴𝑇𝑑𝑖 𝑃𝑗 𝐵𝑤𝑖 𝐵𝑤𝑇𝑖𝑃𝑗𝐵𝑤𝑖 − (1 + 𝛼) 𝑆
∗
𝐴𝑇𝑖 𝑃𝑗𝐶𝑖
]] 𝐴𝑇𝑑𝑖 𝑃𝑗 𝐶𝑖 ]]]] 𝐵𝑤𝑇 𝑖 𝑃𝑗 𝐶𝑖 𝐶𝑖𝑇𝑃𝑗𝐶𝑖 − (1 + 𝛼) 𝑅]
(21)
𝑐12max {𝜆max (𝑃̃𝑖)} + 𝑐12𝑑𝜆max (𝑄̃) + 𝛿2𝜆max (𝑆)
𝑖∈𝑀
+ 𝑐2𝜌2𝜆
(𝑅̃)
<
𝑐22min𝑖∈𝑀 {𝜆min (𝑃̃𝑖)} ,
(22)
2 𝑖 max
(1 + 𝛼)𝑁
where 𝑃̃𝑖 = 𝐺−1/2𝑃𝑖𝐺−1/2, 𝑄̃ = 𝐺−1/2𝑄𝐺−1/2, 𝑅̃ = 𝐺−1/2𝑅𝐺−1/2, and 𝜆max(⋅), 𝜆min(⋅) indicate the maximal and minimal eigenvalues of the augment, respectively.
Proof. For the closed-loop system (19), choose a stochastic Lyapunov function candidate as
𝑘−1
𝑉𝑖 (𝑘) = 𝑥𝑘𝑇𝑃𝑖𝑥𝑘 + ∑ 𝑥𝑓𝑇𝑄𝑥𝑓.
(23)
𝑓=𝑘−𝑑
Simple calculation shows that
𝐸 {𝑉𝑖 (𝑘 + 1)} − 𝑉𝑖 (𝑘) = 𝑥𝑘𝑇 (𝐴𝑇𝑖 𝑃𝑗𝐴𝑖 − 𝑃𝑖 + 𝑄) 𝑥𝑘 + 2𝑥𝑘𝑇𝐴𝑇𝑖 𝑃𝑗𝐴𝑑𝑖𝑥𝑘−𝑑
+ 2𝑥𝑘𝑇𝐴𝑇𝑖 𝑃𝑗𝐵𝑤𝑖𝑤𝑘 + 2𝑥𝑘𝑇𝐴𝑇𝑖 𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) + 𝑥𝑘𝑇−𝑑 (𝐴𝑇𝑑𝑖𝑃𝑗𝐴 𝑑𝑖 − 𝑄) 𝑥𝑘−𝑑 + 2𝑥𝑘𝑇−𝑑𝐴𝑇𝑑𝑖𝑃𝑗𝐵𝑤𝑖𝑤𝑘 + 2𝑥𝑘𝑇−𝑑𝐴𝑇𝑑𝑖𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) + 𝑤𝑘Τ𝐵𝑤𝑇𝑖𝑃𝑗𝐵𝑤𝑖𝑤𝑘 + 2𝑤𝑘𝑇𝐵𝑤𝑇𝑖𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) + Δ𝑓𝑖𝑇 (𝑥𝑘) 𝐶𝑖𝑇𝑃𝑗𝐶𝑖Δ𝑓𝑖 (𝑥𝑘) = 𝜁𝑘𝑇Ω𝑖𝜁𝑘,
(24)
where
𝑠
𝑃𝑗 ≜ ∑𝜋𝑖𝑗𝑃𝑗,
𝑗=1
𝜁𝑘 = [𝑥𝑘𝑇 𝑥𝑘𝑇−𝑑 𝑤𝑘𝑇 Δ𝑓𝑖𝑇 (𝑥𝑘)]𝑇,
𝐴𝑇𝑖 𝑃𝑗𝐴𝑖 − 𝑃𝑖 + 𝑄
Ω𝑖 = [[[
𝐴𝑇𝑑𝑖 𝑃𝑗 𝐴𝑖 𝐵𝑤𝑇 𝑖 𝑃𝑗 𝐴𝑖
[ 𝐶𝑖𝑇𝑃𝑗𝐴𝑖
∗ 𝐴𝑇𝑑𝑖𝑃𝑗𝐴 𝑑𝑖 − 𝑄
𝐵𝑤𝑇𝑖𝑃𝑗𝐴 𝑑𝑖 𝐶𝑖𝑇𝑃𝑗𝐴 𝑑𝑖
∗
∗ 𝐵𝑤𝑇 𝑖 𝑃𝑗 𝐵𝑤𝑖 𝐶𝑖𝑇 𝑃𝑗 𝐵𝑤𝑖
∗ ∗∗ ]]] .
𝐶𝑖𝑇𝑃𝑗𝐶𝑖] (25)
Conditions (21) and (24) imply that
𝐸 {𝑉𝑖 (𝑘 + 1)}
≤ (1 + 𝛼) 𝑥𝑘𝑇𝑃𝑖𝑥𝑘 + (1 + 𝛼) 𝑤𝑘𝑇𝑆𝑤𝑘
𝑘−1
+ (1 + 𝛼) Δ𝑓𝑖𝑇 (𝑥𝑘) 𝑅Δ𝑓𝑖 (𝑥𝑘) + (1 + 𝛼) ∑ 𝑥𝑓𝑇𝑄𝑥𝑓
𝑓=𝑘−𝑑
= (1 + 𝛼) 𝑉𝑖 (𝑘) + (1 + 𝛼) 𝑤𝑘𝑇𝑆𝑤𝑘
+ (1 + 𝛼) Δ𝑓𝑖𝑇 (𝑥𝑘) 𝑅Δ𝑓𝑖 (𝑥𝑘) . (26)
Noting that 𝛼 ≥ 0, we can obtain from (26) that
𝑘
𝑉𝑖 (𝑘) ≤ (1 + 𝛼)𝑘𝑉𝑖 (0) + ∑ (1 + 𝛼)𝑘−𝑓+1𝑤𝑓𝑇−1𝑆𝑤𝑓−1
𝑓=1
𝑘
+ ∑ (1 + 𝛼)𝑘−𝑓+1𝑐22𝜌𝑖2𝜆max (𝑅̃)
𝑓=1
Abstract and Applied Analysis
−1
= (1 + 𝛼)𝑘 [𝑥0𝑇𝑃𝑖𝑥0 + ∑ 𝑥𝑓𝑇𝑄𝑥𝑓
[
𝑓=−𝑑
𝑘
+ ∑ (1 + 𝛼)1−𝑓𝑤𝑓𝑇−1𝑆𝑤𝑓−1
𝑓=1
𝑘
+ ∑ (1 + 𝛼)1−𝑓𝑐22𝜌𝑖2𝜆max (𝑅̃)]
𝑓=1
]
≤ (1 + 𝛼)𝑁 [𝑐12max {𝜆max (𝑃̃𝑖)} + 𝑐12𝑑𝜆max (𝑄̃)
𝑖∈𝑀
+ 𝛿2𝜆max (𝑆) + 𝑐22𝜌𝑖2𝜆max (𝑅̃)] .
(27)
Note that
𝑘−1
𝑉𝑖 (𝑘) = 𝑥𝑘𝑇𝑃𝑖𝑥𝑘 + ∑ 𝑥𝑓𝑇𝑄𝑥𝑓
𝑓=𝑘−𝑑
≥ 𝑥𝑇𝑃 𝑥
(28)
𝑘 𝑖𝑘
≥ min {𝜆min (𝑃̃𝑖)} 𝑥𝑘𝑇𝐺𝑥𝑘.
𝑖∈𝑀
According to (27)-(28), one has
𝑥𝑘𝑇𝐺𝑥𝑘 ≤ ((1 + 𝛼)𝑁 (𝑐12max {𝜆max (𝑃̃𝑖)} + 𝑐12𝑑𝜆max (𝑄̃)
𝑖∈𝑀
+𝛿2𝜆max (𝑆) + 𝑐22𝜌𝑖2𝜆max (𝑅̃) ) )
−1
× (min {𝜆min (𝑃̃𝑖)}) .
𝑖∈𝑀
(29)
Condition (19) implies that, for 𝑘 ∈ {1, 2, . . . , 𝑁}, 𝐸{𝑥𝑘𝑇𝐺𝑥𝑘} < 𝑐22. This completes the proof.
Now, we direct our attention to present a solution to the problem of finite-time stabilizing controller design. Such controller is provided by the following theorem.
Theorem 5. The closed-loop system (19) is stochastic finitetime stabilizable via state feedback with respect to the given (𝑐1, 𝑐2, 𝐺, 𝑁, 𝛿) and scalar 𝛼 ≥ 0, if there exist matrices 𝑋𝑖 = 𝑋𝑖𝑇 > 0, 𝑌𝑖, 𝐻 = 𝐻𝑇 > 0, 𝑆 = 𝑆𝑇 > 0, and 𝑅 = 𝑅𝑇 > 0 and scalars 𝜆1, 𝜆2, 𝜆3, 𝜆4 > 0 such that
− (1 + 𝛼) 𝑋𝑖 [[ 𝑁1𝑖 [[[ 00 [ 𝑋𝑖
< 0,
𝑁1𝑇𝑖 −𝑀5𝑖 + 𝑁5𝑖
𝑀3𝑇𝑖 𝑀4𝑇𝑖
0
0 𝑀3𝑖 − (1 + 𝛼) 𝑆
0 0
0 𝑀4𝑖
0
− (1 + 𝛼) 𝑅 0
𝑋𝑖 0 ]] 00 ]]] −𝐻]
(30)
5
𝜆1𝐺−1 < 𝑋𝑖 < 𝐺−1,
𝜆2𝐺−1 < 𝐻,
𝑆 < 𝜆3𝐼,
𝑅 < 𝜆4𝐺, (31)
[− 𝑐22 + 𝛿2𝜆3 + 𝑐22𝜌𝑖2𝜆4 𝑐1 √𝑑𝑐1]
[ (1 + 𝛼)𝑁
[
𝑐
−𝜆 0 ]] < 0. (32)
1
1
[
√𝑑𝑐1
0 −𝜆2 ]
Proof. By using Schur complement, from condition (21) in Lemma 4, it follows that
− (1 + 𝛼) 𝑃𝑖 + 𝑄 ∗
∗
∗
[[
0
−𝑄 ∗
∗
[[[ 00
0 − (1 + 𝛼) 𝑆 ∗
0
0 − (1 + 𝛼) 𝑅
∗ ∗ ]] ∗∗ ]]] ≤ 0,
[
𝑀1𝑖
𝑀2𝑖 𝑀3𝑖
𝑀4𝑖 −𝑀5𝑖] (33)
where
𝑇
𝑇𝑇
𝑀1𝑖 = [√𝜋𝑖1𝐴𝑖 , . . . , √𝜋𝑖𝑠𝐴𝑖 ] ,
𝑀2𝑖 = [√𝜋𝑖1𝐴𝑇𝑑𝑖, . . . , √𝜋𝑖𝑠𝐴𝑇𝑑𝑖]𝑇,
𝑀3𝑖 = [√𝜋𝑖1𝐵𝑤𝑇𝑖, . . . , √𝜋𝑖𝑠𝐵𝑤𝑇𝑖]𝑇,
(34)
𝑀4𝑖 = [√𝜋𝑖1𝐶𝑖𝑇, . . . , √𝜋𝑖𝑠𝐶𝑖𝑇]𝑇,
𝑀5𝑖 = diag {𝑃1−1, . . . , 𝑃𝑠−1} .
Performing matrix elementary transformation to the above inequality, we have
− (1 + 𝛼) 𝑃𝑖 + 𝑄
[[
𝑀1𝑖
[[[ 00
[
0
𝑀1𝑇𝑖 −𝑀5𝑖 𝑀3𝑇𝑖 𝑀4𝑇𝑖 𝑀2𝑇𝑖
0 𝑀3𝑖 − (1 + 𝛼) 𝑆
0
0
0 𝑀4𝑖
0
− (1 + 𝛼) 𝑅
0
0 𝑀2𝑖]]
00 ]]] ≤ 0.
−𝑄 ] (35)
Performing a congruence to the above condition by diag {𝑃𝑖−1 𝐼 𝐼 𝐼 𝐼}, using Schur complement, and letting 𝑋𝑖 = 𝑃𝑖−1 and 𝑌𝑖 = 𝐾𝑖𝑋𝑖, we get
− (1 + 𝛼) 𝑋𝑖 + 𝑋𝑖𝑄𝑋𝑖 [[ 𝑁01𝑖
[
0
𝑁1𝑇𝑖 −𝑀5𝑖 + 𝑁5𝑖
𝑀3𝑇𝑖 𝑀4𝑇𝑖
0 𝑀3𝑖 − (1 + 𝛼) 𝑆
0
0 𝑀04𝑖 ]]
− (1 + 𝛼) 𝑅]
≤ 0, (36)
6
Abstract and Applied Analysis
where
𝑁1𝑖
=
[√𝜋𝑖1(𝐴̃𝑖𝑋𝑖
+
𝑇
𝐵𝑖𝑌𝑖) ,
.
.
.
,
√𝜋𝑖𝑠(𝐴̃𝑖𝑋𝑖
+
𝑇𝑇
𝐵𝑖𝑌𝑖) ] ,
𝑁5𝑖
𝜋𝑖1 𝐴 𝑑𝑖 𝐻𝐴𝑇𝑑𝑖
√𝜋𝑖1√𝜋𝑖2𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 ⋅ ⋅ ⋅ √𝜋𝑖1√𝜋𝑖𝑠𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖
[[[√𝜋𝑖2√𝜋𝑖1𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 𝜋𝑖2𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 ⋅ ⋅ ⋅ √𝜋𝑖2√𝜋𝑖𝑠𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖]]]
= [[
]] ,
[ ...
... d ... ]
[√𝜋𝑖𝑠√𝜋𝑖1𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 √𝜋𝑖𝑠√𝜋𝑖2𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖
𝜋𝑖𝑠𝐴𝑑𝑖𝐻𝐴𝑇𝑑𝑖 ]
𝐴̃𝑖 = ∑ 𝜇𝜎𝑖 𝐴𝜎𝑖 + 𝐴𝑖.
𝜎𝑖 ∈Θ
(37)
By using Schur complement to (36) and letting 𝐻 = 𝑄−1, we
obtain the linear matrix inequality (30) in Theorem 5.
On the other hand, we consider
𝜆 (𝑋̃ ) = 1 ,
max 𝑖 𝜆min (𝑃̃𝑖)
(38)
𝑋̃𝑖 = 𝑃̃𝑖−1 = 𝐺1/2𝑋𝑖𝐺1/2.
Condition (22) follows that
𝑐12
+ 𝑐2𝑑𝜆 (𝑄̃) + 𝛿2𝜆 (S)
min𝑖∈𝑀 {𝜆min (𝑋̃𝑖)} 1 max
max
(39)
<
𝑐22
.
max𝑖∈𝑀 {𝜆max (𝑋̃𝑖)} (1 + 𝛼)𝑁
It is easy to check that the above inequality is guaranteed by imposing the following conditions
𝜆max (𝑋̃𝑖) < 1, 𝜆1 < 𝜆min (𝑋̃𝑖) ,
𝜆max (𝑄̃) < 𝜆2, 𝜆max (𝑆) < 𝜆3,
(40)
𝑐12 + 𝑐2𝑑𝜆 + 𝛿2𝜆 < 𝑐22
𝜆1 1 2
3 (1 + 𝛼)𝑁
which are equivalent to conditions (31)-(32). This completes the proof.
Remark 6. It is worth pointing out that Theorem 5 is not a delay-dependent sufficient criterion, which is conservative when the delay is small. Delay-dependent result can be developed in the same way by choosing a Lyapunov functional that includes more entries, as was done in [36], or delay fractioning approach that can be employed as was done in [31–33].
Remark 7. The coupling relationship between time delay and given finite-time horizon of the underlying system is obtained through a finite-time stable constraint (32) in Theorem 5. From condition (32), it can be seen that, in given finite-time horizon, if the time delay 𝑑 is larger, constraint (32) is more difficult to be satisfied, which means that the existence of time delay increases the instability of system.
4. Numerical Example
Consider discrete-time Markov jump nonlinear system (1) with three operation modes and the following data:
𝐴1 = [00..8480 −−00..0752] , 𝐴𝑑1 = [−00.2.2 00.1.15] ,
𝐵1 = [21] , 𝐵𝑤1 = [00..45] , 𝐶1 = [00.1] ,
𝐴2 = [0.280 00..2342] ,
𝐴
𝑑2
=
[
−0.6 0.2
00..46] ,
𝐵2 = [−11] , 𝐵𝑤2 = [00..26] , 𝐶2 = [00.3] , (41)
𝐴3 = [−0.08.08 00..1646] , 𝐴𝑑3 = [−00.2.3 00..15] ,
𝐵3 = [11] , 𝐵𝑤2 = [00..13] , 𝐶3 = [00.5] ,
𝑓1 (𝑥𝑘) = 𝑓2 (𝑥𝑘) = 𝑓3 (𝑥𝑘) = sin (𝑥1𝑘) cos (𝑥2𝑘) .
Now, a single hidden layer neural network with 2 hidden
neurons was chosen to approximate the nonlinear functions 𝑓𝑖(𝑥𝑘). All parameters of activation functions associated with the hidden layer were chosen to be 𝑞𝑖𝑙 = 0.5 and 𝜆𝑖𝑙 = 1. For these activation functions, we have 𝑠𝑖𝑙(0, 𝜙𝑖𝑙) = 0 and 𝑠𝑖𝑙(1, 𝜙𝑖𝑙) = 1. The connection weights are trained offline by using BP algorithm. The initial weights and state vector are placed by uniformly distributed random numbers in [−1 1].
After 1000 training steps, the optimal approximation weights
are as follows:
𝑊∗ = [−0.86017 −0.81881] ,
1 −0.95025 0.96405
(42)
𝑊2∗ = [−0.57752 −0.58342] .
The upper bound of approximation error is estimated as 𝜌𝑖 = 0.022. Obviously, in this case, we have Θ = 22 × 21. According to (15), 𝐴𝜎𝑖 can be obtained as follows:
𝐴 𝑖1 = 𝐴 𝑖2 = 𝐴 𝑖3 = 𝐴 𝑖4 = 𝐴 𝑖5 = 𝐴 1⊕[0,0,0]𝑇 = 𝐴0⊕[𝑖,𝑗,𝑘]𝑇 [00 00] , (𝑖, 𝑗, 𝑘 ∈ {0, 1}) ,
𝐴𝑖6 = 𝐴1⊕[1,0]𝑇 = [0.490677 0.470288] ,
(43)
𝐴𝑖7 = 𝐴1⊕[0,1]𝑇 = [0.550439 −0.506245] ,
𝐴𝑖8 = 𝐴1⊕[1,1]𝑇 = [1.00512 −0.0809567] .
The initial state and initial mode are taken as 𝑥0 = [−0.3 0.4]𝑇 and 𝑟0 = 1, respectively. The iterative step is taken as 𝑁 = 7. The mode path from time step 0 to time
Abstract and Applied Analysis
m
3.5
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
Time step
Figure 1: Jump modes.
x2
7
3
2
1
0
−1
−2
−3
−3
−2
−1
0
1
2
3
x1
Figure 3: State trajectory under finite-time control.
x2
15
10
5
0
−5
−10
−15
−5
0
5
10
15
20
25
x1
Figure 2: State trajectory of free MJS.
step 7 is generated randomly and it is shown in Figure 1. Let 𝑐1 = 0.5, 𝑐2 = 2, 𝑁 = 7, 𝐺 = I, 𝑑 = 0.5, 𝛼 = 0.5, and 𝛿2 = 1. By solving the matrix inequalities in Theorem 5, we have the
following controller gains:
𝐾1 = [−0.9304 −0.0683] ,
𝐾2 = [−1.7231 0.3654] ,
(44)
𝐾3 = [1.1486 −0.1588] .
The state trajectories of the free and controlled MJLS (16) are drawn in Figures 2 and 3, respectively. It could be seen that the free MJLS (16) is not stochastic FTB because the trajectory exceeds the given bound 𝑐22. However, the trajectory is limited between the two ellipsoids regions by employing the proposed control move which satisfactorily justify that the closed-loop MJLS (16) is stochastic FTB.
It should be pointed out that, in the simulation example, as long as the choice of initial condition is satisfied with ‖𝑥0𝑇𝑅𝑟𝑥0‖ ≤ 𝑐1, then the system is robustly finite-time stabilizable; that is, system trajectories stay within a given bound.
5. Conclusions
The finite-time stabilization problem for discrete-time Markovian jump nonlinear system with time delay and normbounded exogenous disturbance is investigated in this paper. The nonlinearities are parameterized by multilayer neural network and the relationship between time delay and given finite-time horizon is explored with delay-independent conditions. The proposed framework is versatile and can accommodate a number of challenging design problems including finite-time control and filtering of discrete-time or continuous-time nonlinear MJS with parameter uncertainties, time delays, and so on. The future work can consider some delay-dependent approaches or delay fractioning approaches to reduce the conservativeness introduced by time delay.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (Grant no. 61104121) and the 111 Project (Grant no. B12018), and the third author would also like to thank the Alexander-von-Humboldt Foundation for providing the support to this research.
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