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Quantitative mean square exponential stability and stabilization of stochastic systems with Markovian switching
Accepted Manuscript
Quantitative Mean Square Exponential Stability and Stabilization of Stochastic Systems with Markovian Switching Zhiguo Yan, Yunxia Song, Ju H. Park PII: DOI: Reference:
S0016-0032(18)30150-9 10.1016/j.jfranklin.2018.02.026 FI 3353
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
31 July 2017 11 January 2018 4 February 2018
Please cite this article as: Zhiguo Yan, Yunxia Song, Ju H. Park, Quantitative Mean Square Exponential Stability and Stabilization of Stochastic Systems with Markovian Switching, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.02.026
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Quantitative Mean Square Exponential Stability and Stabilization of Stochastic Systems with Markovian Switching∗ Zhiguo Yana,b , Yunxia Songa , Ju H. Parkb a School
of Electrical Engineering and Automation, Qilu University of Technology, Jinan 250353, PR China. of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea.
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b Department
Abstract
This paper is concerned with the quantitative mean square exponential stability and stabilization for stochastic systems with Markovian switching. First, the concept of quantitative mean square exponential stabil-
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ity(QMSES) is introduced, and two stability criteria are derived. Then, based on an auxiliary definition of general finite-time mean square stability(GFTMSS), the relations among QMSES, GFTMSS and finite time stochastic stability (FTSS) are obtained. Subsequently, QMSE-stabilization is investigated and several new sufficient conditions for the existence of the state and observer-based controllers are provided by means of linear matrix inequalities. An algorithm is given to achieve the relation between the minimum states’ upper bound and the states’ decay velocity. Finally, a numerical example is utilized to show the merit of
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the proposed results.
Keywords: Quantitative mean square exponential stability, stochastic systems, Markovian switching,
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matrix inequality
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1. Introduction
During the past decades, considerable interests have been focused on stochastic systems due to its extensive applications in many practical systems [1]-[11]. Meanwhile, Markovian switching systems have
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been widely used in modeling dynamic systems with random changes in structures and parameters, such as manufacturing systems [12], networked systems [13], fault-tolerant systems [14]. Considering the advantages
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of stochastic systems and Markovian switching systems, much attention has been paid to stochastic systems with Markovian switching, and many results have been reported. For instance, reference [15] investigated the finite-time stability and stabilization problems for stochastic Markovian jump systems. The reference [16] gave pth moment exponential stability conditions for stochastic functional differential equations with Levy noise and Markov switching by using the Razumikhin method and Lyapunov functions. The sliding This work was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant number NRF-2017R1A2B2004671), National Natural Science Foundation of China under Grant nos. 61403221 and Project funded by China Postdoctoral Science Foundation (2017M610425). Email: [email protected](Zhiguo Yan), songyx [email protected](Yunxia Song), [email protected](Ju H. Park). Preprint submitted to Elsevier
March 14, 2018
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mode controller design for singular stochastic Markovian jump systems with uncertainties was considered in [17]. The reference [18] studied the robust H∞ control problems for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method. For more other related results, readers can refer to [23]-[31] and the references therein. Recently, the problems of exponential stability have been studied for various system models and many results have been obtained. For instance, the exponential stability of port-Hamiltonian systems was investi-
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gated via energy-shaped method in [32]. The exponential stability for linear neutral time-varying differential systems was addressed in [33] by spectral properties of Metzler matrices and a comparison principle. The exponential stability of nonlinear differential systems with time-varying delay was discussed in [34]. However, in these existing literatures, the analysis on exponential stability mainly concerns the steady state behavior of systems, and the transient behaviors of systems are not considered. This may result in bad transient
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behaviors and very long setting time for an exponentially stable system, which is not expected in practical systems. The reasons for these problems are mainly from the definition of exponential stability. As is well known, the linear system model
x(t) ˙ = Ax(t), x(0) = x0 ∈ Rn
(1)
is exponentially stable if there exist two constants β>1 and α>0 such that kx(t)k2 <βkx0 k2 e−αt , t≥0. From the above definition, we can see that it only requires the existences of β and α, which can not guarantee a
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satisfactory transient performance and short setting time. In fact, it is very necessary to both have steady performance and satisfactory transient behavior for practical systems. Nevertheless, up to now, there are
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nearly no literatures to make up these defects of exponential stability. Motivated by aforementioned discussions, this paper devotes to investigate the quantitative mean square exponential stability and stabilization problems for stochastic systems with Markovian switching. Because
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of the special structure of this kind of systems, the issues considered are more complicated. By utilizing stochastic analysis technique, the stability criteria and several stabilization conditions are obtained. The
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main contributions of this paper are listed as: 1)The concept of quantitative mean square exponential stability is introduced for stochastic systems with Markov switching, which is a kind of mean square exponential stability with prescribed upper bound and decay rate of the trajectories. 2) Two stability criteria
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are derived, and the relations among QMSES, GFTMSS and FTSS are obtained. 3)The state feedback and observer-based controllers are designed by LMIs, and an algorithm is provided to achieve the relation between the minimum states’ upper bound and states’ decay velocity. This paper is organized as follows: Section 2 presents a quantitative mean square exponential stability wrt (β, α) and its relation to general finite-time mean square stability wrt (β, T ) and FTSS. Section 3 provides several stability conditions for quantitative exponential mean square stability wrt (β, α), general
finite-time mean square stability wrt (β, T ), finite-time stochastic stability and analyzes the relations among 2
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these stability conditions. Section 4 designs state and output feedback QMSE-stabilization controllers. A numerical algorithm is given in Section 5. Section 6 employs an example to illustrate the results. Section 7 gives the conclusion. Notations: X 0 stands for transpose of a matrix X. The X>0 means that X is positive-definite. In×n stands for n×n identity matrix. λmax (X)(λmin (X)) represents the maximum(minimum) eigenvalue of a
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matrix X. E[·] denotes the operator of the mathematical expectation. tr(X) is the trace of a matrix X. The √ √ notation kxk:= x0 x denotes Euclidian 2-norm of a vector x and kxkQ denotes x0 Qx. The asterisk “∗” represents the symmetry term in a matrix and diag{· · · } represents a block-diagonal matrix. The shorthand “wrt” is an abbreviation of “with respect to”.
2. A Quantitative Mean Square Exponential Stability and Its Relations to Finite-time Stochas-
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tic Stability
Let w(t) be a scalar Brownian motion defined on the probability space (Ω, F, Ft , P ). Let rt be a right-continuous Markov chain taking values in a finite state space S={1, 2, · · · , N } and the transition rate
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matrix Π=[γij ]N ×N . We assume that rt is independent of w(t) and has the following transition probability: γ 4t + o(4t), i 6= j, ij P {rt+4t = j|rt = i} = 1 + γ 4t + o(4t), i = j, ii
where 4t > 0, γij is the stationary transition rate from mode i to mode j with γij >0, i6=j and γii =−
P
i6=j
γij .
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Ft stands for the smallest σ-algebra generated by w(s), r(s), 0≤s≤t, i.e., Ft =σ{w(s), r(s)|0≤s≤t}.
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Consider the following linear stochastic Itˆo system with Markovian switching dx(t) = (A(r )x(t) + B(r )u(t))dt + (A(r )x(t) + B(r )u(t))dw(t), t t t t y(t) = C(r )x(t), x(0) = x ∈ Rn , r ∈ S, t
0
(2)
0
where x(t)∈Rn is the state, u(t)∈Rm is the control input and y(t)∈Rp is the measurement output. For
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rt =i, A(rt ), A(rt ), B(rt ), B(rt ) and C(rt ) are constant matrices of compatible dimensions, denoted by Ai , Ai , Bi , B i and Ci for simplicity.
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Next, the definition of quantitative mean square exponential stability for system (2) is provided. Definition 1. Given the constants β≥1 and α>0. The free system (2) (that is, u(t) ≡ 0) is called quanti-
tative mean square exponentially stable wrt (β, α) if for all x0 ∈Rn \{0} and any r0 ∈ S, E[kx(t)k2 |x0 , r0 ] < βkx0 k2 e−αt , t ≥ 0.
(3)
Remark 1. Note that, β and α are pre-specified in Definition 1. The β embodies the upper bound of the free system (2) and α embodies the decay velocity of the free system (2). By Definition 1, the free system (2) will have a satisfactory transient behavior if β is smaller and converge to zero more quickly if α is bigger. 3
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Remark 2. Definition 1 is different from exponential mean square stability (EMS-stability) in [35]. The EMS-stability is as follows: a system is said to be EMS-stable if there exist constants d1 >0 and d2 >0 such that E[kx(t)k2 |x0 , r0 ]≤d1 kx0 k2 e−d2 t , ∀t≥0, for all x0 ∈Rn and r0 ∈S. The EMS-stability only requires the existence of constants d1 >0 and d2 >0, which may result in large transient behavior (if d1 is large) and small decay velocity (if d2 is small) of systems. In other words, the existing d1 and d2 may not be what we
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need in practical applications. In the following, a numerical example is given to show EMS-stability may not guarantee satisfactory transient performance and steady state performance.
Example 1. Consider the free system (2) with two modes and the system parameters are given by " # " # " # " # A1 =
−0.2
0.6
0
−0.05
, A1 =
1
−0.06
0.01
0.1
, A2 =
−0.1
0.8
0
−0.04
, A2 =
0.2
−0.2
0.04
0.6
, x(0) = [−1 1]0 .
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By using the Euler-Maruyama method, one can obtain the curves of kx(t)k2 (10 curves) and E[kx(t)k2 ] of the free system (2) in Figure 1. From Figure 1, it can be seen that the free system (2) is EMS-stable but it has bad transient performance and very small decay velocity. These performances are not what we need in
Figure 1: Responses of the free system (2).
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engineering practice.
In Example 1, the free system is mean square exponential stable, because there exist β = 10000 and
α = 0.1, such that the inequality E[kx(t)k2 |x0 , r0 ] < 10000kx0 k2 e−0.1t , t ≥ 0, holds. But the autonomous system is not quantitative mean square exponential stable with respect to (β = 10, α = 5), because the inequality E[kx(t)k2 |x0 , r0 ] < 10kx0 k2 e−5t , t ≥ 0, does not hold. The relationship between them is that: Quantitative mean square exponential stability with respect to (α, β) implies mean square exponential stability, but the converse does not hold. Next, a proposition equivalent to Definition 1 is given. 4
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Proposition 1. For given the constants β≥1 and α>0, the free system (2) is quantitative mean square exponentially stable wrt (β, α) if and only if N N X X tr( Zi (t)) < β · tr( Zi (0))e−αt , t ≥ 0, i=1
i=1
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where Zi (t)>0 is the solution to N P Z˙ i (t) = Ai Zi (t) + Zi (t)A0i + Ai Zi (t)A0i + γji Zj (t), j=1
Z (0) = x(0)x0 (0)1 i {r0 =i} .
(4)
Proof: Let Zi (t)=E[x(t)x0 (t)1{rt =i} ], i ∈ S, where 1{rt =i} stands for the Dirac measure. Denote Z(t) = (Z1 (t), Z2 (t), · · · , ZN (t)), we obtain N P
i=1
E[kx(t)k2 1{r(t)=i} ] =
According to (5), it can be obtained that
N P
i=1
(E[tr(x(t)x0 (t)1{r(t)=i} )]) =
βkx(0)k2 e−αt = β · tr( From (5) and (6), it can be implied that E[kx(t, x0 , r0 )k ] < βkx0 k e
N X i=1
i=1
tr(Zi (t)) = tr(
N P
Zi (t)).
(5)
i=1
Zi (0))e−αt , t ≥ 0.
(6)
N N X X ⇐⇒ tr( Zi (t)) < β · tr( Zi (0))e−αt , t ≥ 0.
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2 −αt
2
N P
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E[kx(t)k2 ] =
i=1
i=1
Applying generalized Itˆo formula for E[x(t)x (t)1{rt =i} ], it can be obtained that Zi (t) satisfies the ordinary
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0
differential equation (4). The proof is completed.
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Remark 3. Proposition 1 is actually to solve the ordinary differential equation (4), which provides an easier method to test quantitative mean square exponential stability wrt (β, α) of the free system (2).
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Definition 2. Given constants β≥1 and T >0. The free system (2) is general finite-time mean square stable
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wrt (β, T ) if for any x0 ∈Rn \{0}, r0 ∈ S, E[kx(t)k2 |x0 , r0 ] < βkx(0)k2 , ∀t ∈ [0, T ].
Proposition 2. For given constants β≥1 and T >0, the free system (2) is general finite-time mean square stable wrt (β, T ) if and only if N N X X tr( Zi (t)) < β · tr( Zi (0)), ∀t ∈ [0, T ], i=1
i=1
where Zi (t)>0 is the solution to equation (4). In the following, the definition of finite-time stability in [36] is extended to the free system (2). 5
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Definition 3. The free system (2) is said to be finite-time stochastically stable wrt (c1 , c2 , T ), where 0 < c1 < c2 , and T > 0, if kx(0)k2 < c1 ⇒ E[kx(t)k2 ] < c2 , ∀t ∈ [0, T ]. Next, the relations among quantitative mean square exponential stability wrt (β, α), general finite-time mean square stability wrt (β, T ) and finite-time stochastic stability wrt (c1 , c2 , T ) are given in the theorem
Theorem 1. The following relations hold: (i)
QMSES wrt (β, α) =⇒
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below.
GFTMSS wrt (β, T )
(ii)⇓
⇓(iii)
EMS-stability
FTSS wrt (c1 , c2 , T )
(i)
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Proof: [=⇒] From Definition 1, it can be seen that Ekx(t)k2 <βkx0 k2 e−αt ≤βkx0 k2 for t∈[0, T ], so this relation is immediate. (ii)
[=⇒] From the definition of EMS-stability and Definition 1, (ii) is obtained. (iii)
[=⇒] If Ekx0 k2
Remark 4. From the relations (i) and (iii) in Theorem 1, we obtain that if Ekx0 k2
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tive mean square exponential stability wrt (β, α) implies FTSS for t∈[0, T ]. In addition, in quantitative mean square exponential stability wrt (β, α), if there exists a t0 ∈[0, T ] such that βe−αt0 kx0 k2 =c1 holds,
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then E[kx(t)k2 ]≤c1 for t∈[t0 , T ], which shows that the trajectories are contractive for t∈[t0 , T ]. However,
FTS in Definition 3 cannot show this contractiveness of the trajectories for t∈[0, T ], which only requires E[kx(t)k2 ]
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Based on Definition 1, the definition of quantitative mean square exponential stabilization wrt (β, α) can be given as follows.
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Definition 4. Given the constants β>1 and α>0. System (2) is called quantitative mean square exponential
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stabilizable wrt (β, α) if there exists a feedback controller u∗ (t) such that the closed-loop system dx(t) = [A(r )x(t) + B(r )u∗ (t)]dt + [A(r )x(t) + B(r )u∗ (t)]dw(t), t t t t x(0) = x ∈ Rn ,
(7)
0
is quantitative mean square exponentially stable wrt (β, α).
3. A Quantitative Mean Square Exponential Stability Theorem and Its Relations to GFTMSS and FTSS in Stability Conditions This section gives a sufficient condition for QMSES, a sufficient condition for GFTS and a sufficient condition for FTSS are given, respectively. Based on these stability conditions, the relations among QMSES, 6
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GFTMSS, and FTSS are shown. Theorem 2. For given β≥1 and α>0, the free system (2) is quantitative mean square exponentially stable wrt (β, α) if there exists a set of matrices Qi >0, i ∈ S, such that the following inequalities hold: max{λmax (Qi )}/ min{λmin (Qi )} < β, i∈S
(8)
i∈S
N X j=1
γij Qj < −αQi .
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0
A0i Qi + Qi Ai + Ai Qi Ai +
(9)
Proof: For each rt =i, i∈S, construct a stochastic quadratic function as V (t, x(t), i) = eαt x0 (t)Qi x(t), where Qi satisfies (8)-(9). Applying generalized Itˆo formula [35] for V (t, x(t), i), we have
dV (t, x(t), i) = [αeαt x0 (t)Qi x(t) + eαt Lx0 (t)Qi x(t)]dt + 2eαt x0 (t)Qi Ai x(t)dw(t),
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where
0
Lx0 (t)Qi x(t) = x0 (t)[A0i Qi + Qi Ai + Ai Qi Ai +
N X
γij Qj ]x(t).
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we have
Using (9) and (11), one has
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By (13) and (14), we have
Rt 0
αeαs x0 (s)Qrs x(s)ds + E
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E[V (x(t), i)] = E[V (x(0), r0 )] + E
(11)
j=1
Integrating both sides of (10) on [0, t] and then taking expectation and using the equality Z t E 2eαs x0 (s)Qrs Ars x(s)dw(s) = 0, 0
(10)
Rt 0
eαs Lx0 (s)Qrs x(s)ds.
(12)
(13)
Lx0 (t)Qi x(t) < −αx0 (t)Qi x(t).
(14)
E[V (x(t), i)] < E[V (x(0), r0 )].
(15)
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According to the following inequalities
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E[V (x(t), i)] = E[eαt x0 (t)Qi x(t)] ≥ eαt min{λmin (Qi )}Ekx(t)k2 , i∈S
E[V (x(0), r0 )] = E[x0 (0)Qr0 x(0)] ≤ max{λmax (Qi )}Ekx(0)k2 , i∈S
(15) implies
E[kx(t)k2 ] <
max{λmax (Qi )} i∈S
min{λmin (Qi )} i∈S
kx0 k2 e−αt .
(16)
By (8), (16) implies the free system (2) is quantitative mean square exponentially stable wrt (β, α). The proof is completed . In the following, a sufficient condition for general finite-time mean square stability wrt (β, T ) will be given. 7
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Theorem 3. For given β>1 and T >0, the free system (2) is general finite-time mean square stable wrt (β, T ) if there exist a set of matrices Qi >0, i ∈ S, and a nonnegative scalar h such that the following inequalities hold: max{λmax (Qi )}/ min{λmin (Qi )} < βe−hT , i∈S
i∈S
0
A0i Qi + Qi Ai + Ai Qi Ai +
N X
γij Qj < hQi .
(17) (18)
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j=1
Proof: Let Qi >0(i ∈ S) such that (17) and (18) are satisfied. Defining V (x(t), i) = x0 (t)Qi x(t) and using Itˆ o formula [35], we have
dV (x(t), i) = LV (x(t), i)dt + 2x0 (t)Qi Ai x(t)dw(t), where 0
N X
γij Qj ]x(t).
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LV (x(t), i) = x0 (t)[A0i Qi + Qi Ai + Ai Qi Ai +
(19)
(20)
j=1
Integrating both sides of (19) on [0, t], t∈[0, T ] and then taking expectation and using the equality (12), we have
E[V (x(t), i)] = E[V (x(0), r0 )] + E Using (18) and (20), one has
Rt 0
LV (x(s), rs )ds.
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LV (x(t), i) < hV (x(t), i).
(21)
(22)
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By (21), (22) and Gronwall inequality [37], we obtain
E[V (x(t), i)] < ehT E[V (x(0), r0 )].
(23)
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According to the following inequality
E[V (x(t), i)] = E[x0 (t)Qi x(t)] ≥ min{λmin (Qi )}Ekx(t)k2 , i∈S
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E[V (x(0), r0 )] = E[x (t)Qr0 x(t)] ≤ max{λmax (Qi )}Ekx(0)k2 , 0
i∈S
(23) leads to
E[kx(t)k2 ] <
max{λmax (Qi )} i∈S
min{λmin (Qi )}
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i∈S
kx0 k2 ehT .
(24)
By (17), (24) implies that the free system (2) is general finite-time mean square stable wrt (β, T ). The proof is completed. Next, a sufficient condition for FTSS of the free system (2) is presented.
Theorem 4. For given c1 >0 and c2 >0, the free system (2) is finite-time stochastic stable if there exist a set of matrices Qi >0, i ∈ S and a nonnegative scalar h such that the following inequalities hold: max{λmax (Qi )}/ min{λmin (Qi )} < i∈S
i∈S
8
c2 −hT e , c1
(25)
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0
A0i Qi + Qi Ai + Ai Qi Ai +
N X
γij Qj < hQi .
(26)
j=1
Proof: Following the proof of Theorem 3, it can be obtained that E[kx(t)k2 ] <
max{λmax (Qi )} i∈S
min{λmin (Qi )} i∈S
kx0 k2 ehT .
(27)
2
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By the given conditionE[kx(0)k ] < c1 and (25), (27) implies that E[kx(t)k2 ] < c2 . Therefore, this completes the proof.
(28)
The Stability Condition Relation Between QMSES wrt (β, α) and GFTMSS wrt (β, T ): From (8) and (17), we obtain: when h=0, (8) and (9) implies (17) and (18), This shows that the free system
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(2) is general finite-time stable wrt (β, T ).
The Stability Condition Relation Between GFTMSS wrt (β, T ) and FTSS: From (17) and (25), the following relations are obtained: when (2) is FTSS wrt (c1 , c2 , T ).
c2 c1 ≥β,
(17) implies (25). This shows that the free system
4. Quantitative Mean Square Exponential Stabilization
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In this section, we will design a state feedback controller and and an observer-based controller such that
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the resulting closed-loop system (7) is quantitative mean square exponentially stable. 4.1. QMSE-stabilization via State feedback Controller This subsection aims to design the state feedback controller u(t)=K(rt )x(t) such that the closed-loop
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system (7) is quantitative mean square exponentially stable wrt (β, α). K(rt ) is the controller gain and is denoted by Ki for rt = i for simplicity. Next, a sufficient condition for the existence of Ki (i ∈ S) is
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presented.
Theorem 5. For given constants β>1 and α>0, the closed-loop system (7) is quantitative mean square exponentially stable wrt (β, α) if there exist matrices Xi >0, Yi (i ∈ S), and scalars δ1 >0, δ2 >0 such that
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the following matrix inequalities hold: Φ i11 ∗ ∗
0
0
Xi Ai + Yi0 B i
Φi13
−Xi
0
∗
−Φi33
< 0,
δ2 I < Xi < δ1 I, δ1 − βδ2 < 0,
(29)
(30)
√ √ √ √ where Φi11 = Ai Xi +Xi A0i +Bi Yi +Yi0 Bi0 +αXi +γii Xi , Φi13 = [ γi,1 Xi , · · · , γi,i−1 Xi , γi,i+1 Xi , · · · , γi,N Xi ], Φi33 = diag{X1 , · · · , Xi−1 , Xi+1 , · · · , XN }. In this case, Ki = Yi Xi−1 , i = 1, 2, · · · , N . 9
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Proof: Substituting u(t)=Ki x(t) into system (2), we have the closed-loop system dx(t) = (Ai + Bi Ki )x(t)dt + (Ai + B i Ki )x(t)dw(t).
(31)
√ √ √ −1 −1 −1 −1 In inequality (29), letting Xi =Q−1 i , then Yi = Ki Qi , Φi13 = [ γi,1 Qi , · · · , γi,i−1 Qi , · · · , γi,N Qi ], PN −1 −1 −1 −1 Φi33 = diag{Q−1 = Φi13 Φi33 Φ0i13 . By using 1 , · · · , Qi−1 , · · · , QN } and note that j=1,j6=i γij Qi Qj Qi
Schur Complement, (29) is equivalent to the following inequality
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−1 −1 −1 0 0 Q−1 i (Ai + Bi Ki ) + (Ai + Bi Ki )Qi + Qi (Ai + B i Ki ) Qi (Ai + B i Ki )Qi PN −1 + j=1 γij Q−1 < −αQ−1 i Qj Qi i .
(32)
Pre- and post-multiplying (32) by Qi , it becomes the following inequality
(Ai + Bi Ki )0 Qi + Qi (Ai + Bi Ki ) + (Ai + B i Ki )0 Qi (Ai + B i Ki ) +
PN
j=1
γij Qj < −αQi .
(33)
So, the inequality (29) is equivalent to (33). By Theorem 2, the closed-loop system (31) is quantitative mean
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square exponential stable wrt (β, α) if there exists a set of matrices Qi >0(i ∈ S) such that (8) and (33). On the other hand, the first inequality of (30) implies that 1 1 I < Qi < I. δ1 δ2
(34)
Considering (34) and the second inequality of (30), we obtain (8), i.e., max{λmax (Qi )} i∈S
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min{λmin (Qi )}) i∈S
δ1 < β. δ2
ED
This completes the proof.
<
4.2. QMSE-stabilization via Observer-based Controller In above subsection, we have designed a state feedback controller u(t) = K(rt )x(t). However, it is
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difficult to measure the total states in practice, so an observer-based output feedback controller is considered dˆ x(t) = [A(rt )ˆ x(t) + B(rt )u(t) + L(rt )(y(t) − C(rt )ˆ x(t))]dt, u(t) = K(r )ˆ x(t), x ˆ(0) = 0,
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as follows:
(35)
t
where x ˆ(t)∈Rn is the estimation of the state of x(t), and L(rt ) is the estimator gain matrices and is denoted
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by Li (i ∈ S) for rt = i for simplicity. Let e(t)=x(t)−ˆ x(t), the closed-loop system (7) can be written as dx(t) = [(A + B K )x(t) − B K e(t)]dt + [(A + B K )x(t) − B K e(t)]dw(t), i i i i i i i i i i x(0) = x ,
(36)
0
with
de(t) = (A − L C )e(t)dt + [(A + B K )x(t) − B K e(t)]dw(t), i i i i i i i i e(0) = x − x ˆ . 0
0
The following theorem gives a sufficient condition of the existence of Li (i ∈ S). 10
(37)
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Theorem 6. For given the constants β>1 and α>0, the closed-loop system (36) is quantitative mean square exponentially stable wrt (β, α) if there exist matrices Gi >0, Hi >0, Mi , i ∈ S, and scalars θ1 >0, θ2 >0, θ3 >0 such that the following inequalities hold:
Πi11 + αGi
Πi12
∗
Πi22 + αHi
< 0,
(38)
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θ1 I < Gi < θ2 I, 0 < Hi < θ3 I, θ2 + θ3 < βθ1 ,
(39)
where Πi11 = (Ai + Bi Ki )0 Gi + Gi (Ai + Bi Ki ) + (Ai + B i Ki )0 Gi (Ai + B i Ki ) + (Ai + B i Ki )0 Hi (Ai + B i Ki ) + PN 0 0 0 j=1 γij Gj , Πi12 = −Gi Bi Ki − (Ai + B i Ki ) Gi B i Ki − (Ai + B i Ki ) Hi B i Ki , Πi22 = (B i Ki ) Gi B i Ki + PN (B i Ki )0 Hi B i Ki + Hi Ai − Mi Ci + A0i Hi − Ci0 Mi0 + j=1 γij Hj . In this case, a gain is given by Li =Hi−1 Mi , i = 1, 2, · · · , N .
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e i =diag{Gi , Hi } and z(t)=[x0 (t), e0 (t)]0 , where Gi >0, Hi >0 satisfy (38)-(39), we take Proof: Let Q e i z(t) = x0 (t)Gi x(t) + e0 (t)Hi e(t). V (z(t), rt = i) = z 0 (t)Q
Applying Itˆ o formula for V (x(t), i), we have
e(t)
0
Πi11
Πi12
∗
Π∗i22
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LV (z(t), rt = i) =
x(t)
x(t) e(t)
,
where Π∗i22 = (B i Ki )0 Gi B i Ki + (B i Ki )0 Hi B i Ki + (Ai − Li Ci )0 Hi + Hi (Ai − Li Ci ) +
(41) PN
j=1
γij Hj .
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By Schur’s complement and letting Mi =Hi Li , inequality (38) gives LV (z(t), i) < −αV (z(t), i).
(42)
Integrating both sides of (42) on [0, t] and taking expectation yields Z t E[V (z(t), i)] < E[V (z(0), r0 )] − αE V (z(s), rs )ds.
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(40)
(43)
0
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By Gronwall inequality, (43) implies that E[V (z(t), i)] < E[V (z(0), r0 )]e−αt .
(44)
According to given conditions, we have the following inequalities: E[V (z(t), i)] = E[x0 (t)Gi x(t) + e0 (t)Hi e(t)] ≥ min{λmin (Gi )}Ekx(t)k2 , i∈S
EV (z(0), r0 ) = E[x0 (0)Gi x(0) + e0 (0)Hi e(0)] ≤ [max{λmax (Gi )} + max{λmax (Hi )}]Ekx(0)k2 . i∈S
i∈S
(45) (46)
Considering (45) and (46), it obtains that Ekx(t)k2 <
max{λmax (Gi )} + max{λmax (Hi )} i∈S
i∈S
min{λmin (Gi )} i∈S
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kx(0)k2 e−αt .
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From inequalities in (39), we obtain that Ekx(t)k2 < βkx(0)k2 e−αt . Therefore, the closed-loop system (36) is quantitative mean square exponentially stable wrt (β, α). This completes the proof. Remark 5. The design for observer-based controller (34) actually includes two steps. The first step is to design Ki , i = 1, · · · , N by Theorem 5 and the second step is to design Li , i = 1, · · · , N by Theorem 6. The superiority of this design method is convenient for finding the solutions of Ki , i = 1, · · · , N and
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Li , i = 1, · · · , N . 5. Numerical Algorithm
In this section, an algorithm is given to achieve the relation between the minimum value of β and α. The following analysis and algorithm for Theorem 5 are first given. A similar algorithm can be applied to Theorem 6.
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Analysis: By analyzing (29)-(30), we find that if (29)-(30) have feasible solutions wrt (Xi , Yi , δ1 , δ2 ) for given β and α, then βmin is existent and αmin =0. So, we first fix α1 =0 and search for βmin from 1, and then we fix α2 and search for βmin from 1 again, and so forth. The detailed algorithm is as follows. Algorithm 1 Step 1 : Given a step size dα for α and a step size dβ for β, and a large number l.
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Step 2 : Set m=1, n=1 and take α1 =0, β1 =1.
Step 3 : If (αm , βn ) makes (29)-(30) have feasible solutions, store (αm , βn ) into (U (m), V (n)), go to Step 4 ; else if β
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Step 4 : Let αm =αm +dα , m=m+1, then return to Step 3.
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Step 5 : Stop.
By running Algorithm 1, the following results are obtained. 1) If (U (m), V (n))=(0, 0), then we cannot find (α, βmin ) making (29)-(30) have feasible solutions.
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Otherwise, there exists (α, βmin ) making (29)-(30) have feasible solutions. 2) According to (U (m), V (n)), a curve of βmin versus α can be obtained, by which, we also can obtain
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a βmin for a given decay rate α and a αmax for a given β. Remark 6. In Algorithm 1, l is a very large number. If (αm , β=l) can not make (29)-(30) have feasible solutions, then βmin corresponding to αm can not be found. In addition, Algorithm 1 can be used to design feedback controllers for the desired β and α. 6. Simulation Results In this section, an illustrative example is provided to demonstrate the effectiveness of the obtained results. Consider system (2) with parameters as follows: 12
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1
2
Ai −0.15
1
−0.06 −0.2 1 0 −0.04 0
Ai 0.3 −0.05
0.01 0.1 0.2 −0.2 0.04 0.6
h Let the scalars β=4 and α=0.3, x(0)= −1
1
i0
Bi
0.8
1
0.6
0.5
Bi
0.3
0.2
1
0.6
Ci
−0.1
−0.2
−0.3 −0.5
0
0
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i
. Using Euler-Maruyama method to simulate the
standard Brownian motion, one can obtain the curves of kx(t)k2 (10 curves) and E[kx(t)k2 ] of the free system (2) in Figure 2. From Figure 2, we see that the free system (2) is EMS-stable but it has bad transient performance and very small decay velocity, and it is not quantitative mean square exponentially stable for given β=4 and α=0.3. So, it is necessary to design feedback controllers such that the closed-loop
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system has the desired transient performance and steady state performance.
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Figure 2: Responses of the free system (2).
6.1. State-feedback QMSE-stabilization wrt (β, α)
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For given β=4 and α=0.3, by solving (29) and (30) in Theorem 5, the solutions are as follows: h i h i K1 = −0.4009 −0.6103 , K2 = 0.0660 −0.7000 .
Figure 3 shows that the closed-loop system (31) is (β, α)-stable. Applying Algorithm 1, the relation
curve between βmin and α is obtained, which is illustrated by Figure 4. According to Figure 4, the maximum value of α, i.e., the maximum decay, is 2.37 for β=4. When β≥24, the maximum decay is 2.8.
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Figure 3: Responses of closed-loop system (31).
Figure 4: βmin versus α for state feedback case
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6.2. Observer-based QMSE-stabilization wrt (β, α) Based on state feedback case, the feedback controller u(t)=Ki x ˆ(t) is chosen. By solving (38) and (39)
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in Theorem 6, we obtain:
h i0 h i0 L1 = −5.6373 −8.5821 , L2 = −4.2532 −17.6682 .
Figure 5 shows that the closed-loop system (36) is (β, α)-stable. Applying Algorithm 1, the relation
curve between βmin and α is obtained, which is illustrated by Figure 6. According to Figure 6, the maximum value of α, i.e., the maximum decay, is 0.52 when β≥10.8. Remark 7. Comparing Figure 2 with Figure 3 and Figure 5, we see that quantitative mean square exponential stability wrt (β, α) guarantees the desired transient performance and decay velocity of the system. 14
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Figure 5: Responses of closed-loop system (36).
Figure 6: βmin versus α for observer-based case.
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On the other hand, from Figure 4 and Figure 6, we see that the biggest decay rate can be found for the
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desired transient performance.
7. Conclusion
In this paper, QMSES and QMSES-stabilization for stochastic systems with Markovian switching have
been studied. The relations among QMSES, GFTS and FTSS have been obtained. Furthermore, the state feedback and observer-based controllers have been designed for this class of systems, respectively. Finally, the relations between βmin and α has been given by a numerical algorithm. In the future work, quantitative mean square exponential stability can be also extended to stochastic neural networks [38]-[40]. 15
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Quantitative Mean Square Exponential Stability and Stabilization of Stochastic Systems with Markovian Switching Zhiguo Yan, Yunxia Song, Ju H. Park PII: DOI: Reference:
S0016-0032(18)30150-9 10.1016/j.jfranklin.2018.02.026 FI 3353
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
31 July 2017 11 January 2018 4 February 2018
Please cite this article as: Zhiguo Yan, Yunxia Song, Ju H. Park, Quantitative Mean Square Exponential Stability and Stabilization of Stochastic Systems with Markovian Switching, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.02.026
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Quantitative Mean Square Exponential Stability and Stabilization of Stochastic Systems with Markovian Switching∗ Zhiguo Yana,b , Yunxia Songa , Ju H. Parkb a School
of Electrical Engineering and Automation, Qilu University of Technology, Jinan 250353, PR China. of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea.
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b Department
Abstract
This paper is concerned with the quantitative mean square exponential stability and stabilization for stochastic systems with Markovian switching. First, the concept of quantitative mean square exponential stabil-
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ity(QMSES) is introduced, and two stability criteria are derived. Then, based on an auxiliary definition of general finite-time mean square stability(GFTMSS), the relations among QMSES, GFTMSS and finite time stochastic stability (FTSS) are obtained. Subsequently, QMSE-stabilization is investigated and several new sufficient conditions for the existence of the state and observer-based controllers are provided by means of linear matrix inequalities. An algorithm is given to achieve the relation between the minimum states’ upper bound and the states’ decay velocity. Finally, a numerical example is utilized to show the merit of
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the proposed results.
Keywords: Quantitative mean square exponential stability, stochastic systems, Markovian switching,
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matrix inequality
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1. Introduction
During the past decades, considerable interests have been focused on stochastic systems due to its extensive applications in many practical systems [1]-[11]. Meanwhile, Markovian switching systems have
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been widely used in modeling dynamic systems with random changes in structures and parameters, such as manufacturing systems [12], networked systems [13], fault-tolerant systems [14]. Considering the advantages
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of stochastic systems and Markovian switching systems, much attention has been paid to stochastic systems with Markovian switching, and many results have been reported. For instance, reference [15] investigated the finite-time stability and stabilization problems for stochastic Markovian jump systems. The reference [16] gave pth moment exponential stability conditions for stochastic functional differential equations with Levy noise and Markov switching by using the Razumikhin method and Lyapunov functions. The sliding This work was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant number NRF-2017R1A2B2004671), National Natural Science Foundation of China under Grant nos. 61403221 and Project funded by China Postdoctoral Science Foundation (2017M610425). Email: [email protected](Zhiguo Yan), songyx [email protected](Yunxia Song), [email protected](Ju H. Park). Preprint submitted to Elsevier
March 14, 2018
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mode controller design for singular stochastic Markovian jump systems with uncertainties was considered in [17]. The reference [18] studied the robust H∞ control problems for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method. For more other related results, readers can refer to [23]-[31] and the references therein. Recently, the problems of exponential stability have been studied for various system models and many results have been obtained. For instance, the exponential stability of port-Hamiltonian systems was investi-
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gated via energy-shaped method in [32]. The exponential stability for linear neutral time-varying differential systems was addressed in [33] by spectral properties of Metzler matrices and a comparison principle. The exponential stability of nonlinear differential systems with time-varying delay was discussed in [34]. However, in these existing literatures, the analysis on exponential stability mainly concerns the steady state behavior of systems, and the transient behaviors of systems are not considered. This may result in bad transient
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behaviors and very long setting time for an exponentially stable system, which is not expected in practical systems. The reasons for these problems are mainly from the definition of exponential stability. As is well known, the linear system model
x(t) ˙ = Ax(t), x(0) = x0 ∈ Rn
(1)
is exponentially stable if there exist two constants β>1 and α>0 such that kx(t)k2 <βkx0 k2 e−αt , t≥0. From the above definition, we can see that it only requires the existences of β and α, which can not guarantee a
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satisfactory transient performance and short setting time. In fact, it is very necessary to both have steady performance and satisfactory transient behavior for practical systems. Nevertheless, up to now, there are
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nearly no literatures to make up these defects of exponential stability. Motivated by aforementioned discussions, this paper devotes to investigate the quantitative mean square exponential stability and stabilization problems for stochastic systems with Markovian switching. Because
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of the special structure of this kind of systems, the issues considered are more complicated. By utilizing stochastic analysis technique, the stability criteria and several stabilization conditions are obtained. The
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main contributions of this paper are listed as: 1)The concept of quantitative mean square exponential stability is introduced for stochastic systems with Markov switching, which is a kind of mean square exponential stability with prescribed upper bound and decay rate of the trajectories. 2) Two stability criteria
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are derived, and the relations among QMSES, GFTMSS and FTSS are obtained. 3)The state feedback and observer-based controllers are designed by LMIs, and an algorithm is provided to achieve the relation between the minimum states’ upper bound and states’ decay velocity. This paper is organized as follows: Section 2 presents a quantitative mean square exponential stability wrt (β, α) and its relation to general finite-time mean square stability wrt (β, T ) and FTSS. Section 3 provides several stability conditions for quantitative exponential mean square stability wrt (β, α), general
finite-time mean square stability wrt (β, T ), finite-time stochastic stability and analyzes the relations among 2
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these stability conditions. Section 4 designs state and output feedback QMSE-stabilization controllers. A numerical algorithm is given in Section 5. Section 6 employs an example to illustrate the results. Section 7 gives the conclusion. Notations: X 0 stands for transpose of a matrix X. The X>0 means that X is positive-definite. In×n stands for n×n identity matrix. λmax (X)(λmin (X)) represents the maximum(minimum) eigenvalue of a
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matrix X. E[·] denotes the operator of the mathematical expectation. tr(X) is the trace of a matrix X. The √ √ notation kxk:= x0 x denotes Euclidian 2-norm of a vector x and kxkQ denotes x0 Qx. The asterisk “∗” represents the symmetry term in a matrix and diag{· · · } represents a block-diagonal matrix. The shorthand “wrt” is an abbreviation of “with respect to”.
2. A Quantitative Mean Square Exponential Stability and Its Relations to Finite-time Stochas-
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tic Stability
Let w(t) be a scalar Brownian motion defined on the probability space (Ω, F, Ft , P ). Let rt be a right-continuous Markov chain taking values in a finite state space S={1, 2, · · · , N } and the transition rate
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matrix Π=[γij ]N ×N . We assume that rt is independent of w(t) and has the following transition probability: γ 4t + o(4t), i 6= j, ij P {rt+4t = j|rt = i} = 1 + γ 4t + o(4t), i = j, ii
where 4t > 0, γij is the stationary transition rate from mode i to mode j with γij >0, i6=j and γii =−
P
i6=j
γij .
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Ft stands for the smallest σ-algebra generated by w(s), r(s), 0≤s≤t, i.e., Ft =σ{w(s), r(s)|0≤s≤t}.
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Consider the following linear stochastic Itˆo system with Markovian switching dx(t) = (A(r )x(t) + B(r )u(t))dt + (A(r )x(t) + B(r )u(t))dw(t), t t t t y(t) = C(r )x(t), x(0) = x ∈ Rn , r ∈ S, t
0
(2)
0
where x(t)∈Rn is the state, u(t)∈Rm is the control input and y(t)∈Rp is the measurement output. For
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rt =i, A(rt ), A(rt ), B(rt ), B(rt ) and C(rt ) are constant matrices of compatible dimensions, denoted by Ai , Ai , Bi , B i and Ci for simplicity.
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Next, the definition of quantitative mean square exponential stability for system (2) is provided. Definition 1. Given the constants β≥1 and α>0. The free system (2) (that is, u(t) ≡ 0) is called quanti-
tative mean square exponentially stable wrt (β, α) if for all x0 ∈Rn \{0} and any r0 ∈ S, E[kx(t)k2 |x0 , r0 ] < βkx0 k2 e−αt , t ≥ 0.
(3)
Remark 1. Note that, β and α are pre-specified in Definition 1. The β embodies the upper bound of the free system (2) and α embodies the decay velocity of the free system (2). By Definition 1, the free system (2) will have a satisfactory transient behavior if β is smaller and converge to zero more quickly if α is bigger. 3
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Remark 2. Definition 1 is different from exponential mean square stability (EMS-stability) in [35]. The EMS-stability is as follows: a system is said to be EMS-stable if there exist constants d1 >0 and d2 >0 such that E[kx(t)k2 |x0 , r0 ]≤d1 kx0 k2 e−d2 t , ∀t≥0, for all x0 ∈Rn and r0 ∈S. The EMS-stability only requires the existence of constants d1 >0 and d2 >0, which may result in large transient behavior (if d1 is large) and small decay velocity (if d2 is small) of systems. In other words, the existing d1 and d2 may not be what we
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need in practical applications. In the following, a numerical example is given to show EMS-stability may not guarantee satisfactory transient performance and steady state performance.
Example 1. Consider the free system (2) with two modes and the system parameters are given by " # " # " # " # A1 =
−0.2
0.6
0
−0.05
, A1 =
1
−0.06
0.01
0.1
, A2 =
−0.1
0.8
0
−0.04
, A2 =
0.2
−0.2
0.04
0.6
, x(0) = [−1 1]0 .
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By using the Euler-Maruyama method, one can obtain the curves of kx(t)k2 (10 curves) and E[kx(t)k2 ] of the free system (2) in Figure 1. From Figure 1, it can be seen that the free system (2) is EMS-stable but it has bad transient performance and very small decay velocity. These performances are not what we need in
Figure 1: Responses of the free system (2).
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engineering practice.
In Example 1, the free system is mean square exponential stable, because there exist β = 10000 and
α = 0.1, such that the inequality E[kx(t)k2 |x0 , r0 ] < 10000kx0 k2 e−0.1t , t ≥ 0, holds. But the autonomous system is not quantitative mean square exponential stable with respect to (β = 10, α = 5), because the inequality E[kx(t)k2 |x0 , r0 ] < 10kx0 k2 e−5t , t ≥ 0, does not hold. The relationship between them is that: Quantitative mean square exponential stability with respect to (α, β) implies mean square exponential stability, but the converse does not hold. Next, a proposition equivalent to Definition 1 is given. 4
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Proposition 1. For given the constants β≥1 and α>0, the free system (2) is quantitative mean square exponentially stable wrt (β, α) if and only if N N X X tr( Zi (t)) < β · tr( Zi (0))e−αt , t ≥ 0, i=1
i=1
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where Zi (t)>0 is the solution to N P Z˙ i (t) = Ai Zi (t) + Zi (t)A0i + Ai Zi (t)A0i + γji Zj (t), j=1
Z (0) = x(0)x0 (0)1 i {r0 =i} .
(4)
Proof: Let Zi (t)=E[x(t)x0 (t)1{rt =i} ], i ∈ S, where 1{rt =i} stands for the Dirac measure. Denote Z(t) = (Z1 (t), Z2 (t), · · · , ZN (t)), we obtain N P
i=1
E[kx(t)k2 1{r(t)=i} ] =
According to (5), it can be obtained that
N P
i=1
(E[tr(x(t)x0 (t)1{r(t)=i} )]) =
βkx(0)k2 e−αt = β · tr( From (5) and (6), it can be implied that E[kx(t, x0 , r0 )k ] < βkx0 k e
N X i=1
i=1
tr(Zi (t)) = tr(
N P
Zi (t)).
(5)
i=1
Zi (0))e−αt , t ≥ 0.
(6)
N N X X ⇐⇒ tr( Zi (t)) < β · tr( Zi (0))e−αt , t ≥ 0.
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2 −αt
2
N P
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E[kx(t)k2 ] =
i=1
i=1
Applying generalized Itˆo formula for E[x(t)x (t)1{rt =i} ], it can be obtained that Zi (t) satisfies the ordinary
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differential equation (4). The proof is completed.
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Remark 3. Proposition 1 is actually to solve the ordinary differential equation (4), which provides an easier method to test quantitative mean square exponential stability wrt (β, α) of the free system (2).
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Definition 2. Given constants β≥1 and T >0. The free system (2) is general finite-time mean square stable
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wrt (β, T ) if for any x0 ∈Rn \{0}, r0 ∈ S, E[kx(t)k2 |x0 , r0 ] < βkx(0)k2 , ∀t ∈ [0, T ].
Proposition 2. For given constants β≥1 and T >0, the free system (2) is general finite-time mean square stable wrt (β, T ) if and only if N N X X tr( Zi (t)) < β · tr( Zi (0)), ∀t ∈ [0, T ], i=1
i=1
where Zi (t)>0 is the solution to equation (4). In the following, the definition of finite-time stability in [36] is extended to the free system (2). 5
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Definition 3. The free system (2) is said to be finite-time stochastically stable wrt (c1 , c2 , T ), where 0 < c1 < c2 , and T > 0, if kx(0)k2 < c1 ⇒ E[kx(t)k2 ] < c2 , ∀t ∈ [0, T ]. Next, the relations among quantitative mean square exponential stability wrt (β, α), general finite-time mean square stability wrt (β, T ) and finite-time stochastic stability wrt (c1 , c2 , T ) are given in the theorem
Theorem 1. The following relations hold: (i)
QMSES wrt (β, α) =⇒
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below.
GFTMSS wrt (β, T )
(ii)⇓
⇓(iii)
EMS-stability
FTSS wrt (c1 , c2 , T )
(i)
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Proof: [=⇒] From Definition 1, it can be seen that Ekx(t)k2 <βkx0 k2 e−αt ≤βkx0 k2 for t∈[0, T ], so this relation is immediate. (ii)
[=⇒] From the definition of EMS-stability and Definition 1, (ii) is obtained. (iii)
[=⇒] If Ekx0 k2
Remark 4. From the relations (i) and (iii) in Theorem 1, we obtain that if Ekx0 k2
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tive mean square exponential stability wrt (β, α) implies FTSS for t∈[0, T ]. In addition, in quantitative mean square exponential stability wrt (β, α), if there exists a t0 ∈[0, T ] such that βe−αt0 kx0 k2 =c1 holds,
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then E[kx(t)k2 ]≤c1 for t∈[t0 , T ], which shows that the trajectories are contractive for t∈[t0 , T ]. However,
FTS in Definition 3 cannot show this contractiveness of the trajectories for t∈[0, T ], which only requires E[kx(t)k2 ]
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Based on Definition 1, the definition of quantitative mean square exponential stabilization wrt (β, α) can be given as follows.
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Definition 4. Given the constants β>1 and α>0. System (2) is called quantitative mean square exponential
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stabilizable wrt (β, α) if there exists a feedback controller u∗ (t) such that the closed-loop system dx(t) = [A(r )x(t) + B(r )u∗ (t)]dt + [A(r )x(t) + B(r )u∗ (t)]dw(t), t t t t x(0) = x ∈ Rn ,
(7)
0
is quantitative mean square exponentially stable wrt (β, α).
3. A Quantitative Mean Square Exponential Stability Theorem and Its Relations to GFTMSS and FTSS in Stability Conditions This section gives a sufficient condition for QMSES, a sufficient condition for GFTS and a sufficient condition for FTSS are given, respectively. Based on these stability conditions, the relations among QMSES, 6
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GFTMSS, and FTSS are shown. Theorem 2. For given β≥1 and α>0, the free system (2) is quantitative mean square exponentially stable wrt (β, α) if there exists a set of matrices Qi >0, i ∈ S, such that the following inequalities hold: max{λmax (Qi )}/ min{λmin (Qi )} < β, i∈S
(8)
i∈S
N X j=1
γij Qj < −αQi .
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0
A0i Qi + Qi Ai + Ai Qi Ai +
(9)
Proof: For each rt =i, i∈S, construct a stochastic quadratic function as V (t, x(t), i) = eαt x0 (t)Qi x(t), where Qi satisfies (8)-(9). Applying generalized Itˆo formula [35] for V (t, x(t), i), we have
dV (t, x(t), i) = [αeαt x0 (t)Qi x(t) + eαt Lx0 (t)Qi x(t)]dt + 2eαt x0 (t)Qi Ai x(t)dw(t),
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where
0
Lx0 (t)Qi x(t) = x0 (t)[A0i Qi + Qi Ai + Ai Qi Ai +
N X
γij Qj ]x(t).
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we have
Using (9) and (11), one has
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By (13) and (14), we have
Rt 0
αeαs x0 (s)Qrs x(s)ds + E
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E[V (x(t), i)] = E[V (x(0), r0 )] + E
(11)
j=1
Integrating both sides of (10) on [0, t] and then taking expectation and using the equality Z t E 2eαs x0 (s)Qrs Ars x(s)dw(s) = 0, 0
(10)
Rt 0
eαs Lx0 (s)Qrs x(s)ds.
(12)
(13)
Lx0 (t)Qi x(t) < −αx0 (t)Qi x(t).
(14)
E[V (x(t), i)] < E[V (x(0), r0 )].
(15)
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According to the following inequalities
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E[V (x(t), i)] = E[eαt x0 (t)Qi x(t)] ≥ eαt min{λmin (Qi )}Ekx(t)k2 , i∈S
E[V (x(0), r0 )] = E[x0 (0)Qr0 x(0)] ≤ max{λmax (Qi )}Ekx(0)k2 , i∈S
(15) implies
E[kx(t)k2 ] <
max{λmax (Qi )} i∈S
min{λmin (Qi )} i∈S
kx0 k2 e−αt .
(16)
By (8), (16) implies the free system (2) is quantitative mean square exponentially stable wrt (β, α). The proof is completed . In the following, a sufficient condition for general finite-time mean square stability wrt (β, T ) will be given. 7
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Theorem 3. For given β>1 and T >0, the free system (2) is general finite-time mean square stable wrt (β, T ) if there exist a set of matrices Qi >0, i ∈ S, and a nonnegative scalar h such that the following inequalities hold: max{λmax (Qi )}/ min{λmin (Qi )} < βe−hT , i∈S
i∈S
0
A0i Qi + Qi Ai + Ai Qi Ai +
N X
γij Qj < hQi .
(17) (18)
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j=1
Proof: Let Qi >0(i ∈ S) such that (17) and (18) are satisfied. Defining V (x(t), i) = x0 (t)Qi x(t) and using Itˆ o formula [35], we have
dV (x(t), i) = LV (x(t), i)dt + 2x0 (t)Qi Ai x(t)dw(t), where 0
N X
γij Qj ]x(t).
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LV (x(t), i) = x0 (t)[A0i Qi + Qi Ai + Ai Qi Ai +
(19)
(20)
j=1
Integrating both sides of (19) on [0, t], t∈[0, T ] and then taking expectation and using the equality (12), we have
E[V (x(t), i)] = E[V (x(0), r0 )] + E Using (18) and (20), one has
Rt 0
LV (x(s), rs )ds.
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LV (x(t), i) < hV (x(t), i).
(21)
(22)
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By (21), (22) and Gronwall inequality [37], we obtain
E[V (x(t), i)] < ehT E[V (x(0), r0 )].
(23)
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According to the following inequality
E[V (x(t), i)] = E[x0 (t)Qi x(t)] ≥ min{λmin (Qi )}Ekx(t)k2 , i∈S
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E[V (x(0), r0 )] = E[x (t)Qr0 x(t)] ≤ max{λmax (Qi )}Ekx(0)k2 , 0
i∈S
(23) leads to
E[kx(t)k2 ] <
max{λmax (Qi )} i∈S
min{λmin (Qi )}
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i∈S
kx0 k2 ehT .
(24)
By (17), (24) implies that the free system (2) is general finite-time mean square stable wrt (β, T ). The proof is completed. Next, a sufficient condition for FTSS of the free system (2) is presented.
Theorem 4. For given c1 >0 and c2 >0, the free system (2) is finite-time stochastic stable if there exist a set of matrices Qi >0, i ∈ S and a nonnegative scalar h such that the following inequalities hold: max{λmax (Qi )}/ min{λmin (Qi )} < i∈S
i∈S
8
c2 −hT e , c1
(25)
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0
A0i Qi + Qi Ai + Ai Qi Ai +
N X
γij Qj < hQi .
(26)
j=1
Proof: Following the proof of Theorem 3, it can be obtained that E[kx(t)k2 ] <
max{λmax (Qi )} i∈S
min{λmin (Qi )} i∈S
kx0 k2 ehT .
(27)
2
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By the given conditionE[kx(0)k ] < c1 and (25), (27) implies that E[kx(t)k2 ] < c2 . Therefore, this completes the proof.
(28)
The Stability Condition Relation Between QMSES wrt (β, α) and GFTMSS wrt (β, T ): From (8) and (17), we obtain: when h=0, (8) and (9) implies (17) and (18), This shows that the free system
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(2) is general finite-time stable wrt (β, T ).
The Stability Condition Relation Between GFTMSS wrt (β, T ) and FTSS: From (17) and (25), the following relations are obtained: when (2) is FTSS wrt (c1 , c2 , T ).
c2 c1 ≥β,
(17) implies (25). This shows that the free system
4. Quantitative Mean Square Exponential Stabilization
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In this section, we will design a state feedback controller and and an observer-based controller such that
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the resulting closed-loop system (7) is quantitative mean square exponentially stable. 4.1. QMSE-stabilization via State feedback Controller This subsection aims to design the state feedback controller u(t)=K(rt )x(t) such that the closed-loop
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system (7) is quantitative mean square exponentially stable wrt (β, α). K(rt ) is the controller gain and is denoted by Ki for rt = i for simplicity. Next, a sufficient condition for the existence of Ki (i ∈ S) is
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presented.
Theorem 5. For given constants β>1 and α>0, the closed-loop system (7) is quantitative mean square exponentially stable wrt (β, α) if there exist matrices Xi >0, Yi (i ∈ S), and scalars δ1 >0, δ2 >0 such that
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the following matrix inequalities hold: Φ i11 ∗ ∗
0
0
Xi Ai + Yi0 B i
Φi13
−Xi
0
∗
−Φi33
< 0,
δ2 I < Xi < δ1 I, δ1 − βδ2 < 0,
(29)
(30)
√ √ √ √ where Φi11 = Ai Xi +Xi A0i +Bi Yi +Yi0 Bi0 +αXi +γii Xi , Φi13 = [ γi,1 Xi , · · · , γi,i−1 Xi , γi,i+1 Xi , · · · , γi,N Xi ], Φi33 = diag{X1 , · · · , Xi−1 , Xi+1 , · · · , XN }. In this case, Ki = Yi Xi−1 , i = 1, 2, · · · , N . 9
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Proof: Substituting u(t)=Ki x(t) into system (2), we have the closed-loop system dx(t) = (Ai + Bi Ki )x(t)dt + (Ai + B i Ki )x(t)dw(t).
(31)
√ √ √ −1 −1 −1 −1 In inequality (29), letting Xi =Q−1 i , then Yi = Ki Qi , Φi13 = [ γi,1 Qi , · · · , γi,i−1 Qi , · · · , γi,N Qi ], PN −1 −1 −1 −1 Φi33 = diag{Q−1 = Φi13 Φi33 Φ0i13 . By using 1 , · · · , Qi−1 , · · · , QN } and note that j=1,j6=i γij Qi Qj Qi
Schur Complement, (29) is equivalent to the following inequality
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−1 −1 −1 0 0 Q−1 i (Ai + Bi Ki ) + (Ai + Bi Ki )Qi + Qi (Ai + B i Ki ) Qi (Ai + B i Ki )Qi PN −1 + j=1 γij Q−1 < −αQ−1 i Qj Qi i .
(32)
Pre- and post-multiplying (32) by Qi , it becomes the following inequality
(Ai + Bi Ki )0 Qi + Qi (Ai + Bi Ki ) + (Ai + B i Ki )0 Qi (Ai + B i Ki ) +
PN
j=1
γij Qj < −αQi .
(33)
So, the inequality (29) is equivalent to (33). By Theorem 2, the closed-loop system (31) is quantitative mean
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square exponential stable wrt (β, α) if there exists a set of matrices Qi >0(i ∈ S) such that (8) and (33). On the other hand, the first inequality of (30) implies that 1 1 I < Qi < I. δ1 δ2
(34)
Considering (34) and the second inequality of (30), we obtain (8), i.e., max{λmax (Qi )} i∈S
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min{λmin (Qi )}) i∈S
δ1 < β. δ2
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This completes the proof.
<
4.2. QMSE-stabilization via Observer-based Controller In above subsection, we have designed a state feedback controller u(t) = K(rt )x(t). However, it is
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difficult to measure the total states in practice, so an observer-based output feedback controller is considered dˆ x(t) = [A(rt )ˆ x(t) + B(rt )u(t) + L(rt )(y(t) − C(rt )ˆ x(t))]dt, u(t) = K(r )ˆ x(t), x ˆ(0) = 0,
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as follows:
(35)
t
where x ˆ(t)∈Rn is the estimation of the state of x(t), and L(rt ) is the estimator gain matrices and is denoted
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by Li (i ∈ S) for rt = i for simplicity. Let e(t)=x(t)−ˆ x(t), the closed-loop system (7) can be written as dx(t) = [(A + B K )x(t) − B K e(t)]dt + [(A + B K )x(t) − B K e(t)]dw(t), i i i i i i i i i i x(0) = x ,
(36)
0
with
de(t) = (A − L C )e(t)dt + [(A + B K )x(t) − B K e(t)]dw(t), i i i i i i i i e(0) = x − x ˆ . 0
0
The following theorem gives a sufficient condition of the existence of Li (i ∈ S). 10
(37)
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Theorem 6. For given the constants β>1 and α>0, the closed-loop system (36) is quantitative mean square exponentially stable wrt (β, α) if there exist matrices Gi >0, Hi >0, Mi , i ∈ S, and scalars θ1 >0, θ2 >0, θ3 >0 such that the following inequalities hold:
Πi11 + αGi
Πi12
∗
Πi22 + αHi
< 0,
(38)
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θ1 I < Gi < θ2 I, 0 < Hi < θ3 I, θ2 + θ3 < βθ1 ,
(39)
where Πi11 = (Ai + Bi Ki )0 Gi + Gi (Ai + Bi Ki ) + (Ai + B i Ki )0 Gi (Ai + B i Ki ) + (Ai + B i Ki )0 Hi (Ai + B i Ki ) + PN 0 0 0 j=1 γij Gj , Πi12 = −Gi Bi Ki − (Ai + B i Ki ) Gi B i Ki − (Ai + B i Ki ) Hi B i Ki , Πi22 = (B i Ki ) Gi B i Ki + PN (B i Ki )0 Hi B i Ki + Hi Ai − Mi Ci + A0i Hi − Ci0 Mi0 + j=1 γij Hj . In this case, a gain is given by Li =Hi−1 Mi , i = 1, 2, · · · , N .
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e i =diag{Gi , Hi } and z(t)=[x0 (t), e0 (t)]0 , where Gi >0, Hi >0 satisfy (38)-(39), we take Proof: Let Q e i z(t) = x0 (t)Gi x(t) + e0 (t)Hi e(t). V (z(t), rt = i) = z 0 (t)Q
Applying Itˆ o formula for V (x(t), i), we have
e(t)
0
Πi11
Πi12
∗
Π∗i22
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LV (z(t), rt = i) =
x(t)
x(t) e(t)
,
where Π∗i22 = (B i Ki )0 Gi B i Ki + (B i Ki )0 Hi B i Ki + (Ai − Li Ci )0 Hi + Hi (Ai − Li Ci ) +
(41) PN
j=1
γij Hj .
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By Schur’s complement and letting Mi =Hi Li , inequality (38) gives LV (z(t), i) < −αV (z(t), i).
(42)
Integrating both sides of (42) on [0, t] and taking expectation yields Z t E[V (z(t), i)] < E[V (z(0), r0 )] − αE V (z(s), rs )ds.
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(40)
(43)
0
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By Gronwall inequality, (43) implies that E[V (z(t), i)] < E[V (z(0), r0 )]e−αt .
(44)
According to given conditions, we have the following inequalities: E[V (z(t), i)] = E[x0 (t)Gi x(t) + e0 (t)Hi e(t)] ≥ min{λmin (Gi )}Ekx(t)k2 , i∈S
EV (z(0), r0 ) = E[x0 (0)Gi x(0) + e0 (0)Hi e(0)] ≤ [max{λmax (Gi )} + max{λmax (Hi )}]Ekx(0)k2 . i∈S
i∈S
(45) (46)
Considering (45) and (46), it obtains that Ekx(t)k2 <
max{λmax (Gi )} + max{λmax (Hi )} i∈S
i∈S
min{λmin (Gi )} i∈S
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kx(0)k2 e−αt .
(47)
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From inequalities in (39), we obtain that Ekx(t)k2 < βkx(0)k2 e−αt . Therefore, the closed-loop system (36) is quantitative mean square exponentially stable wrt (β, α). This completes the proof. Remark 5. The design for observer-based controller (34) actually includes two steps. The first step is to design Ki , i = 1, · · · , N by Theorem 5 and the second step is to design Li , i = 1, · · · , N by Theorem 6. The superiority of this design method is convenient for finding the solutions of Ki , i = 1, · · · , N and
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Li , i = 1, · · · , N . 5. Numerical Algorithm
In this section, an algorithm is given to achieve the relation between the minimum value of β and α. The following analysis and algorithm for Theorem 5 are first given. A similar algorithm can be applied to Theorem 6.
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Analysis: By analyzing (29)-(30), we find that if (29)-(30) have feasible solutions wrt (Xi , Yi , δ1 , δ2 ) for given β and α, then βmin is existent and αmin =0. So, we first fix α1 =0 and search for βmin from 1, and then we fix α2 and search for βmin from 1 again, and so forth. The detailed algorithm is as follows. Algorithm 1 Step 1 : Given a step size dα for α and a step size dβ for β, and a large number l.
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Step 2 : Set m=1, n=1 and take α1 =0, β1 =1.
Step 3 : If (αm , βn ) makes (29)-(30) have feasible solutions, store (αm , βn ) into (U (m), V (n)), go to Step 4 ; else if β
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Step 4 : Let αm =αm +dα , m=m+1, then return to Step 3.
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Step 5 : Stop.
By running Algorithm 1, the following results are obtained. 1) If (U (m), V (n))=(0, 0), then we cannot find (α, βmin ) making (29)-(30) have feasible solutions.
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Otherwise, there exists (α, βmin ) making (29)-(30) have feasible solutions. 2) According to (U (m), V (n)), a curve of βmin versus α can be obtained, by which, we also can obtain
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a βmin for a given decay rate α and a αmax for a given β. Remark 6. In Algorithm 1, l is a very large number. If (αm , β=l) can not make (29)-(30) have feasible solutions, then βmin corresponding to αm can not be found. In addition, Algorithm 1 can be used to design feedback controllers for the desired β and α. 6. Simulation Results In this section, an illustrative example is provided to demonstrate the effectiveness of the obtained results. Consider system (2) with parameters as follows: 12
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1
2
Ai −0.15
1
−0.06 −0.2 1 0 −0.04 0
Ai 0.3 −0.05
0.01 0.1 0.2 −0.2 0.04 0.6
h Let the scalars β=4 and α=0.3, x(0)= −1
1
i0
Bi
0.8
1
0.6
0.5
Bi
0.3
0.2
1
0.6
Ci
−0.1
−0.2
−0.3 −0.5
0
0
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i
. Using Euler-Maruyama method to simulate the
standard Brownian motion, one can obtain the curves of kx(t)k2 (10 curves) and E[kx(t)k2 ] of the free system (2) in Figure 2. From Figure 2, we see that the free system (2) is EMS-stable but it has bad transient performance and very small decay velocity, and it is not quantitative mean square exponentially stable for given β=4 and α=0.3. So, it is necessary to design feedback controllers such that the closed-loop
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system has the desired transient performance and steady state performance.
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Figure 2: Responses of the free system (2).
6.1. State-feedback QMSE-stabilization wrt (β, α)
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For given β=4 and α=0.3, by solving (29) and (30) in Theorem 5, the solutions are as follows: h i h i K1 = −0.4009 −0.6103 , K2 = 0.0660 −0.7000 .
Figure 3 shows that the closed-loop system (31) is (β, α)-stable. Applying Algorithm 1, the relation
curve between βmin and α is obtained, which is illustrated by Figure 4. According to Figure 4, the maximum value of α, i.e., the maximum decay, is 2.37 for β=4. When β≥24, the maximum decay is 2.8.
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Figure 3: Responses of closed-loop system (31).
Figure 4: βmin versus α for state feedback case
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6.2. Observer-based QMSE-stabilization wrt (β, α) Based on state feedback case, the feedback controller u(t)=Ki x ˆ(t) is chosen. By solving (38) and (39)
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in Theorem 6, we obtain:
h i0 h i0 L1 = −5.6373 −8.5821 , L2 = −4.2532 −17.6682 .
Figure 5 shows that the closed-loop system (36) is (β, α)-stable. Applying Algorithm 1, the relation
curve between βmin and α is obtained, which is illustrated by Figure 6. According to Figure 6, the maximum value of α, i.e., the maximum decay, is 0.52 when β≥10.8. Remark 7. Comparing Figure 2 with Figure 3 and Figure 5, we see that quantitative mean square exponential stability wrt (β, α) guarantees the desired transient performance and decay velocity of the system. 14
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Figure 5: Responses of closed-loop system (36).
Figure 6: βmin versus α for observer-based case.
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On the other hand, from Figure 4 and Figure 6, we see that the biggest decay rate can be found for the
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desired transient performance.
7. Conclusion
In this paper, QMSES and QMSES-stabilization for stochastic systems with Markovian switching have
been studied. The relations among QMSES, GFTS and FTSS have been obtained. Furthermore, the state feedback and observer-based controllers have been designed for this class of systems, respectively. Finally, the relations between βmin and α has been given by a numerical algorithm. In the future work, quantitative mean square exponential stability can be also extended to stochastic neural networks [38]-[40]. 15
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