New results on H∞ filtering for Markov jump systems with uncertain transition rates

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New results on H1 filtering for Markov jump systems with uncertain transition rates$ Hui Liu a, Yucai Ding a,n, Jun Cheng b a b

School of Science, Southwest University of Science and Technology, Mianyang 621010, China School of Science, Hubei University for Nationalities, Enshi, Hubei 445000, China

art ic l e i nf o

a b s t r a c t

Article history: Received 28 October 2016 Received in revised form 25 March 2017 Accepted 17 April 2017

In this paper, we study the H∞ filtering problem for a class of continuous-time Markov jump systems with time-varying uncertainties in transition rates, in which the uncertain transition rates are assumed to be affine parameter-dependent uncertainty models. By converting the affine parameter-dependent uncertainty models for transition rates into time-varying polytopic ones and using the Lyapunov function approach, a sufficient condition on the existence of an H∞ filter is obtained in terms of a parameterdependent matrix inequality. Also, the parameter-dependent matrix inequality is converted into a set of parameter-free linear matrix inequalities which can be solved numerically. Illustrative examples are given to demonstrate the effectiveness and advantages of the approach. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: H1 filtering Markov jump systems Time-varying uncertain transition rates Parameter-dependent Lyapunov function

1. Introduction A switching system is a dynamical hybrid system that consists of a finite number of subsystems and a switching rule governing the switching between subsystems. The overall system behavior depends not only on the dynamics of each subsystem but also on the properties of switching signals. During the past decades, a great number of results on stability analysis and control design problems of switching systems have been reported, see for example [1–3]. In recent years, considerable attention has been paid to the investigation of a special class of switching systems, that is, Markov jump systems (MJSs). This is mainly due to the fact that MJSs can be employed to model the abrupt phenomena such as random failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes. Numerous theoretical results on stochastic stability, stabilization and control of such systems have been developed. See, for example, [4–15] and references therein. Also, some useful applications of MJSs can be found in [15–18]. The switching rules between subsystems of MJSs are governed by transition rates or transition probabilities, which play an ☆ This work was supported by A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department (16ZA0146), the Doctoral Research Foundation of Southwest University of Science and Technology (13zx7141), and the National Natural Science Foundation of China (11501474). n Corresponding author. E-mail address: [email protected] (Y. Ding).

important role in stability analysis and control synthesis since they can affect the behavior and performance of jump systems. For MJSs with completely known transition probabilities, a variety of results have been obtained [19–21]. In [22], it is revealed that the uncertainties in transition rates may lead to instability or at least degraded performance of a system, and a robust state-feedback controller is designed such that the closed-loop system is mean square stable for all admissible uncertainties in transition rates. For further results on analysis and synthesis of MJSs with uncertain transition rates or transition probabilities, we refer the reader to [23]. Based on the consideration that the transition probabilities of MJSs may not be known in most cases, a new concept of MJSs with partially unknown transition rates and some innovative research techniques for such class of systems are proposed in [24,25]. Since their publications, this field has seen a large amount of activities and new results, for instance [26–29]. It should be noted that Markov chain in the aforementioned literature are all homogeneous, which may result in certain limitation in some real-world problems [30,31]. Thus, for some dynamic systems with random abrupt changes, the switching between different modes can be better governed by nonhomogeneous Markov chain than homogeneous ones. Under the assumption of polytopic-type transition probabilities, some results on stability analysis and control synthesis of nonhomogeneous MJSs have been obtained [32–34]. State feedback and many other situations rely on the availability of the internal states. In reality, however, accessibility to the internal states is often impractical, or even impossible. Hence, it is

http://dx.doi.org/10.1016/j.isatra.2017.04.015 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Liu H, et al. New results on H1 filtering for Markov jump systems with uncertain transition rates. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.04.015i

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2

necessary to design an effective filter to estimate the system states from the system's input and output. In the last few years, there has been an increasing attention placed on the filter design for different kinds of MJSs, such as MJSs with uncertain transition rates [35], MJSs with partially unknown transition probabilities [36,37], nonhomogeneous MJSs [38], etc. It is worth pointing out that a time-varying uncertainty in transition rates was provided in [39], that is, πij(θ (t )) = πij + π˜ij(θ (t )), where π˜ij(θ (t )) is the uncertainties in jumping rates πij. However, in the synthesis problems of MJSs with such class of transition rates, the uncertainty parts were almost universally assumed to be norm-bounded, i.e., |π˜ij(θ (t ))| ≤ π˜ij . In addition, the authors of Ref. [40] investigated the H∞ filtering problem for MJSs with polytopic-type uncertainty in transition rates. It is clear that only the upper bounds on uncertainty parts (for norm-bounded uncertainties [39]) or the vertices of polytope (for polytopic-type uncertainties [40]) are used during the filter design. When time-varying uncertainties in transition rates are considered, the above results may bring some conservativeness since the time varying information of transition rates are not fully utilized. To the best of the authors' knowledge, the problem of filtering for MJSs with time-varying uncertain transition rates has not been fully investigated and it is a very challenging problem for investigation, which inspires us for this study. In this paper, the H∞ filtering problem for a class of continuoustime MJSs with time-varying uncertainties in transition probabilities is investigated. By converting the affine parameter-dependent uncertainty models for transition rates into time-varying polytopic ones and using a parameter-dependent Lyapunov function, we will develop a sufficient condition for the existence of an H∞ filter in terms of a parameter-dependent matrix inequality. Furthermore, we will convert the parameter-dependent matrix inequality into a set of parameter-free linear matrix inequalities (LMIs) which can be easily solved by using the Matlab LMI toolbox. Finally, two illustrative examples are given to demonstrate the effectiveness and advantages of the approach. Notation: Throughout this paper, n denotes the n-dimensional Euclidean space; AT represents the transpose of A; The symbol ′*′ in matrix inequality denotes the symmetric term of the matrix; X > 0( < 0) means X is a symmetric positive (negative) definite matrix; {·} refers to the expectation operator with respect to some probability measure; diag (X1 ⋯ Xn) denotes a diagonal matrix, I is used to denote the identity matrix with appropriate dimensions.

2. Problem statement and preliminaries

⎧ λ ijΔ + o(Δ), j≠i Pr{r (t + Δ) = j|r (t ) = i} = ⎨ ⎪ ⎩ 1 + λ iiΔ + o(Δ), j = i ⎪

where Δ > 0 and limΔ → 0

o(Δ) Δ

(2)

= 0; λ ij ≥ 0 for i ≠ j is the transition σ

rate from mode i to mode j and λ ii = − ∑ j = 1, j ≠ i λ ij . In this paper, however, we assume that the transition rates are not precisely known a prior, and they include time-varying uncertainties, that is, λ (θ (t ))≔λ + λ^ (θ (t )), where λ^ (θ (t )) are the uncertainties in ij

ij

ij

ij

transition rates

λij. Without loss of generality, the time-varying

(l )  uncertainties can be described by λ˜ij(θ (t )) = ∑l = 1 θl(t )λ^ij , where σ ∑ λ^ = 0 and θ (t ) ∈ [ θ , θ ]. Thus, the transition rates with timej = 1 ij

l

l

l

varying uncertainties can be rewritten as: 

λ ij(θ (t )) = λ ij +

(l )

∑ θl(t )λ^ij

.

(3)

l= 1

The corresponding transition rate matrix can be denoted as (l ) (l ) (l )  Λ(θ (t )) = Λ + ∑l = 1 θl(t )Λ^ , where Λ = [λ ij ]σ × σ and Λ^ = [λ^ij ]σ × σ . It is clear that the transition rates matrix is an affine parameters-dependent matrix. For convenience in our later discussion, we convert the affine parameter-dependent matrices into polytopic parameter ones. For example, for a single-parameter case, the transition rate matrix with time-varying uncertainty is (1) Λ(θ (t )) = Λ + θ1(t )Λ^ ,

θ1(t ) ∈ [ θ1,

θ1].

(4)

It can be converted into the following polytopic parameter-dependent matrix

Π (α(t )) = α1(t )Π (1) + α2(t )Π (2), where Π

(1)

, Π

Π (1) = Λ( θ1),

(2)

α1(t ) + α2(t ) = 1

(5)

, α1(t ) and α2(t ) are determined by

Π (2) = Λ(θ1),

α1(t ) =

θ1 − θ1(t ) , θ1 − θ1

α2(t ) =

θ1(t ) − θ1 . θ1 − θ1

For the multi-parameter case, the transition rate matrix Λ(θ (t )) can be converted into a more general form as follows: M

Π (α(t )) =

M

∑ αl(t )Π (l), ∑ αl(t ) = 1, αl(t ) ≥ 0 l= 1

l= 1

(6)

(l )

where Π (l = 1, 2, … , M ) represent the vertices of the polyhedron. In this paper, we assume that the jump parameters are nonaccessible for the filter implementation. Therefore, it is desirable to design the following filter:

⎧ xḟ (t ) = Af xf (t ) + Bf y(t ) ⎨ ⎪ ⎩ zf (t ) = Cf xf (t ) ⎪

A continuous-time MJS is described by the following stochastic equations:

⎧ ẋ(t ) = A(r (t ))x(t ) + B(r (t ))ω(t ) ⎪ ⎨ y(t ) = C (r (t ))x(t ) + D(r (t ))ω(t ) ⎪ ⎩ z(t ) = L(r (t ))x(t )

(1)

where x(t ) ∈ n is the state vector, ω(t ) ∈ nω is the external disturbance, y(t ) ∈ ny is the measurement output, and z (t ) ∈ nz is the signal to be estimated. The jumping process {r (t ) , t ≥ 0} is a continuous-time, discrete-state Markov process on the probability space, and takes values in a finite set  = {1, 2, … , σ}. In each mode i ∈  , the matrices Ai ≔A(r (t ) = i ), Bi≔B(r (t ) = i ), Ci≔C (r (t ) = i ), Di≔D(r (t ) = i ) and L i≔L(r (t ) = i ) are known real constant matrices with appropriate dimensions. Generally speaking, when the transition rates are precisely known a prior, the switching between modes is governed by the transition rates

(7)

where xf (t ) ∈ n , zf (t ) ∈ nz , xf (0) = 0, the matrices Af, Bf and Cf are to be determined. T Let us denote x¯ (t ) = ⎡⎣ xT (t ) xTf (t )⎤⎦ and e(t ) = z (t ) − zf (t ). Then, combining (1) and (7) together leads to the error dynamic

⎧ ¯ (̇ t ) = A¯ i xf (t ) + B¯iω(t ) ⎪ x ⎨ ⎪ ⎩ e(t ) = C¯ ix¯ (t )

(8)

where

⎡ A 0⎤ i ⎥, A¯ i = ⎢ ⎢⎣ Bf Ci Af ⎥⎦

⎡ Bi ⎤ ⎥, B¯i = ⎢ ⎢⎣ Bf Di ⎥⎦

C¯ i = ⎡⎣ L i −Cf ⎤⎦.

The purpose of this paper is to derive some LMI conditions for the existence of a solution to the H∞ filtering. To this end, we require the following assumptions, definitions and lemma.

Please cite this article as: Liu H, et al. New results on H1 filtering for Markov jump systems with uncertain transition rates. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.04.015i

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Assumption 1. In the following sections, the parameter-dependent matrix lies in a convex bounded polyhedral domain with M vertices, that is, the parameter-dependent matrix  (α(t )) can M

M

be expressed as  (α(t )) = ∑l = 1 αl(t ) (l ), where ∑l = 1 αl(t ) = 1 and

αl(t ) ≥ 0. Assumption 2. The elements of the parameter vector α(t ) vary slowly with the time, and the variation rate satisfy:

−υl ≤ αl̇ (t ) ≤ υl , υl ≥ 0, l = 1, 2, …, M − 1.

(9)

Definition 1. The error dynamic (8) with ω(t ) = 0 is said to be stochastically stable if, for any initial state (x¯ (0) , r (0) , α(0)),

lim  {∥ x¯ (t )∥2 |(x¯ (0), r (0), α(0))} = 0.

Definition 2. [4] For a given real number γ > 0, system (7) is said to be an H∞ filter of system (1), if the filtering error dynamic (8) is stochastically stable and guarantees a prescribed H∞ disturbance attenuation level γ under the zero initial condition, that is, ∥ z (t )∥E2 < γ ∥ ω(t )∥2 for all nonzero ω(t ) ∈ L2[0 ∞), where

∥ z(t )∥E2

∫0

∞

T

 V (x¯ (t ) , r (t ) , α (t ))=2x¯ ̇ (t )Pi (α (t ))x¯ (t ) ⎞ ⎛ σ + x¯ T (t )⎜ ∑ π ij (α (t ))Pj (α (t ))⎟x¯ (t ) + x¯ T (t )Pi̇ (α (t ))x¯ (t ) ⎟ ⎜ ⎠ ⎝ j =1 ⎛ ⎞ σ ⎜ T = x¯ (t ) Pi (α (t ))A i + A iT Pi (α (t )) + ∑ π ij (α (t ))Pj (α (t )) + Pi̇ (α (t ))⎟x¯ (t ) . ⎜ ⎟ j =1 ⎝ ⎠ (13)

It follows from (10) that i(α(t ) , α̇(t )) < 0. This implies  V (x¯ (t ) , r (t ) , α(t )) < 0. Therefore,  V (x¯ (t ) , r (t ) , α(t )) ≤ − ε ∥ x¯ (t )∥2, where ε = mini ∈  ( λ min( − i(α(t ) , α̇(t )))). Furthermore, for any t > 0, by Dynkin's formula, we have

 {V (x¯ (t ), r (t ), α(t ))} − {V (x¯ (0), r (0), α(0))} ⎧ t ⎫ ≤ − ε ⎨ ∥ x¯ (s )∥2 ds|(x¯ (0), r (0), α(0))⎬ ⎩ 0 ⎭

∫

t →+∞

⎧ ⎪⎡ = ⎨ ⎢ ⎣ ⎪ ⎩

3

1⎫ ⎤2 ⎪ ⎡ z (t )z(t )dt ⎥ ⎬ , ∥ ω(t )∥2 = ⎢ ⎦ ⎪ ⎣ ⎭ T

∫0

∞

1

⎤2 ω (t )ω(t )dt ⎥ . ⎦ T

Lemma 1. (Finsler's lemma) Letting that η ∈ n , Ψ = Ψ T ∈ n × n and  ∈ m × n such that rank( ) = r < n , then the following statements are equivalent:

(14)

which yields ⎧ ⎨ ⎩

∫0

⎫ 1 ∥ x¯ (s )∥2 ds|(x¯ (0) , r (0) , α(0))⎬ ≤ {V (x¯ (0) , r (0) , α(0))} < ∞. ⎭ ε

t

(15)

2

Inequality (15) means that limt →+∞ {∥ x¯ (t )∥ |(x¯ (0) , r (0) , α(0))} = 0. It follows from Definition 1 that the error dynamic (8) with ω(t ) = 0 is stochastically stable. We next show the H∞ disturbance attenuation performance. Set

Jτ = 

τ

{∫

0

}

⎡ zT (t )z(t ) − γ 2ωT (t )ω(t )⎤dt . ⎣ ⎦

(a) ηT Ψη < 0, for all η ≠ 0,  η = 0;

Then, under zero initial condition, we have

(b) ∃  ∈ n × m such that Ψ +  +  T T < 0.

Jτ = 

{∫

τ

0

(16)

}

⎡ zT (t )z(t ) − γ 2ωT (t )ω(t ) +  V (x¯ (t ), r (t ), α(t ))⎤dt ⎣ ⎦

−  { V (x¯ (t ), r (t ), α(t ))} 3. Main results

≤

3.1. H∞ performance analysis

Theorem 1. For a given scalar γ > 0, the error dynamic (8) is stochastically stable with an H∞ disturbance attenuation level γ if there exist differentiable matrix functions Pi(α(t )) > 0, i ∈  , such that

⎡ ¯ ¯ T⎤ ⎢ i(α(t ), α̇(t )) Pi(α(t ))Bi Ci ⎥ ⎢ 0 ⎥⎥ < 0 * −γ 2I ⎢ ⎣ * * −I ⎦

(10)

with

i(α(t ), α̇(t )) = Pi(α(t ))A¯ i + A¯ i Pi(α(t )) +

τ

}

ξ T (t )Ω(α(t ), α̇(t ))ξ (t )dt

(17)

ξ T (t )=⎡⎣ x¯ T (t ), ωT (t )⎤⎦, ⎡ ⎤ T  (α(t ), α̇(t )) + C¯ i (α(t ))C¯ i(α(t )) Pi(α(t ))B¯i(α(t ))⎥ . Ω(α(t ), α̇(t ))=⎢ i ⎢ ⎥ * −γ 2I ⎣ ⎦ Direct application of Schur complement to LMI (10) yields Ω(α(t ) , α̇(t )) < 0, which implies Jτ < 0 for any nonzero ω(t ) and τ > 0. Thus, ∥ z (t )∥E2 < γ ∥ ω(t )∥2. From Definition 2, the error dynamic (8) is stochastically stable with an H∞ disturbance at□ tenuation level γ. This completes the proof. 3.2. H∞ filter design

N

∑ πij(α(t ))Pj(α(t )) + Pi̇ (α(t )) j=1

holds for all admissible values of the parameter vector α(t ) and of its time derivative α̇(t ). Proof. Consider the following parameter-dependent Lyapunov function:

V (x¯ (t ), r (t ), α(t )) = x¯ T (t )P (r (t ), α(t ))x¯ (t ).

0

where

In this subsection, we will study the H∞ performance for the error dynamic (8) with transition rate matrix (6).

T

{∫

(11)

Let  be the weak infinitesimal generator of the random process { x¯ (t ) , r (t ) , α(t )} and it is defined as 1 {  {V (x¯ (t + ▵), r (t + ▵), α(t + ▵))|(x¯ (t ), r (t ), α(t ))} ▵ − V (x¯ (t ), r (t ), α(t ))}. (12)

 V (x¯ (t ), r (t ), α(t )) = lim+ ▵→ 0

We first show the error dynamic (8) with ω(t ) = 0 is stochastically stable. When ω(t ) = 0, it can be verified that

The purpose of this subsection is to provide a sufficient condition for the existence of the H∞ filter. Theorem 2. For a given scalar γ > 0, the error dynamic (8) is stochastically stable with an H∞ disturbance attenuation level

γ if there

exist matrix functions i(α(t )) =  Ti (α(t )) ∈ 2n × 2n , i ∈  , and matrices of appropriate dimensions R, S, W, X, Y, such that ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎤ ⎥ ⎥ π ii(α(t ))i(α(t ))  i 0 0 ⎥ ⎥ − ̇ i(α(t )) ⎥ 2 ⎥<0 * −γ I 0 0 ⎥ * 0 −I 0 ⎥ * * * −diag(i(α(t ))) ⎥ ⎥ ⎥⎦ ˜ i(α(t ))

+  T  Ti −  Ti (α(t )) *

* * *

0

 Ti

 T i(α(t ))

(18)

Please cite this article as: Liu H, et al. New results on H1 filtering for Markov jump systems with uncertain transition rates. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.04.015i

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4

⎡ I −I ⎤⎡ R 0 ⎤ M=⎢ ⎥⎢ ⎥, ⎣ 0 V −T ⎦⎣ S U ⎦

where ⎡ T T ⎤ ⎡ R A i R A iT ST + XT + C iT Y T ⎥ 0⎤ T ,  =⎢ ⎥,  i = ⎢ ⎣ S + W W⎦ T ⎢⎣ 0 ⎥⎦ X ⎡ RBi ⎤ ⎥ , i = ⎡⎣ L i − Z −Z ⎤⎦ ,  i=⎢ ⎢⎣ SBi + YDi ⎥⎦

⎡ RT ST ⎤ ⎥. =⎢ ⎣ 0 UT ⎦

It can be verified that

⎡ I 0⎤ M −T  = ⎢ . ⎣ V V ⎥⎦

i(α(t ))=⎡⎣ π i1(α(t ))I π i2(α(t ))I ⋯ π i(i − 1)(α(t ))I π i(i + 1)(α(t ))I ⋯ π iσ (α(t ))I ⎤⎦ ,

(22)

(

diag(i(α(t )))=diag π i1(α(t ))I π i2(α(t ))I ⋯ π i(i − 1)(α(t ))I π i(i + 1)(α(t ))I ⋯ π iσ (α(t ))I ,

˜ Define ˜ = diag(  ⋯ ) and  = diag M−T  M−T  I I M

˜ i(α(t ))=diag( 1(α(t ))  2(α(t )) ⋯ i − 1(α(t )) i + 1(α(t )) ⋯ σ(α(t )))

follows from (21) that, for any nonzero vector ξ,

holds for all admissible values of the parameter vector α(t ) and of its time derivative α̇(t ). Moreover, for any given nonsingular matrix V, a suitable filter in the form of (7) is given by

ξ T ΦiT (α(t )) −T  T Υi (α(t ), α̇(t )) −1Φi(α(t ))ξ < 0.

(

)

T

Af = VW −1XV −1, Bf = VW −1Y , Cf = ZV −1.

(19)

Proof. Define the transformation matrix Υi(α(t ) , α̇(t )) as follows:

⎡ 0 0 I ⎢ −1 ( α ( )) 0 0 N t M ⎢ i Φi(α(t )) = ⎢ 0 I 0 ⎢ 0 0 I ⎢ ⎢⎣ 0 0 0 ⎡ T T T ¯ 0 C¯ i ⎢ 0 Ai M ⎢ ^ ⎢ * Ni(α(t )) MBi 0 α̇(t )) = ⎢ * * − γI 0 ⎢ * * * −I ⎢ ⎢* * * * ⎣

Φi(α(t )) and matrix

0⎤ ⎥ 0⎥ 0⎥⎥, Υi (α(t ), 0⎥ I ⎥⎦

)

˜ . It

(23)

−1

Φi(α(t ))ξ = η(t ), we can rewrite inequality (23) as

T

η (t ) Υi (α(t ), α̇(t ))η(t ) < 0.

(24)

Taking Hi(α(t )) = ⎡⎣ M − Ni(α(t )) 0 0 0⎤⎦, we have Hi(α(t ))Φi(α(t )) = 0 , which yields Hi(α(t ))η(t ) = 0. It follows from Lemma 1 that, for each mode i, there exists a matrix i such that

 T Υi (α(t ), α̇(t )) + i Hi(α(t )) +  T HiT (α(t ))iT < 0. Without loss of generality, we take i =  T i , T i = ⎣⎡ I 0 0 0 0⎤⎦ . Thus, inequality (25) becomes

(

(25) where

)

 T Υi (α(t ), α̇(t )) + iHi(α(t )) + HiT (α(t ))iT 

⎤ ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ 0 ⎥ −diag(i(α(t )))N˜i(α(t ))⎥⎦ ˜ i(α(t ))M

Set 

−T

T

⎡ ¯ T −1 ¯ T T −1 ˜⎤ ⎢ Ω11 Ω¯ 12 0  M Ci  M i(α(t )) ⎥ ⎥ ⎢ T ¯ ¯ 0 0 ⎥ ⎢ * Ω22  Bi =⎢ * ⎥<0 * − γI 0 0 ⎥ ⎢ * * −I 0 ⎥ ⎢ * ⎥⎦ ⎢⎣ * * * * −Ω¯ 55

(26)

where where

˜ =diag( M M ⋯ M ), N^ (α(t )) = π (α(t ))N (α(t )) − Ṅ (α(t )), M i ii i i

Ω¯ 11 =  T M −1 +  T M −T ,

Ni(α(t )) = NiT (α(t )) > 0, N˜i(α(t ))=diag( N1(α(t )) N2(α(t )) ⋯ Ni − 1(α(t )) Ni + 1(α(t )) ⋯Nσ (α(t ))). Pre- and post- multiplying Υi(α(t ) , α̇(t )) by we have

ΦiT (α(t ))

T Ω¯ 12 =  T M −1A¯ i  −  T M −1Ni(α(t ))M −T ,

Ω¯ 22 = πii(α(t )) T M −1Ni(α(t ))M −T  −  T M −1Ni̇ (α(t ))M −T ,

and Φi(α(t )),

ΦiT (α(t ))Υi (α(t ), α̇(t ))Φi(α(t )) ⎤ ⎡ T −T ˜T ¯ ¯T i(α(t ))M ⎥ ⎢ Ω11 M Ni (α(t ))MBi Ci ⎥ ⎢ * − γI 0 0 =⎢ ⎥ * * − I 0 ⎥ ⎢ ⎢⎣ * * * −diag(i(α(t )))N˜i(α(t ))⎥⎦

Taking into account equality (22) and the definition of  , we have T T  T M −T  =  ,  T M −1C¯ i = ⎡⎣ L − Cf V − Cf V ⎤⎦ ,

⎡ A T RT A T ST + V T AT V −T WT + C T BT V −T WT ⎤ i i f i f T ⎥,  T M −1A¯ i  = ⎢ ⎥ ⎢ 0 T T −T T V Af V W ⎦ ⎣

(20)

where T Ω11= MT Ni−T (α(t ))MA¯ i + A¯ i MT Ni−1(α(t ))M + πii(α(t ))MT Ni−1(α(t ))M T

−M

T ˜ −1 ˜ −T ˜ . diag(i(α(t )))N˜i(α(t ))M Ω¯ 55 = ˜ M

⎤ ⎡ RBi ⎥.  T B¯i = ⎢ −1 ⎢⎣ SBi + WV Bf Di ⎥⎦ Furthermore, set

Ni−T (α(t ))Ni̇ (α(t ))Ni−1(α(t ))M

 T M −1diag(i(α(t )))Ni(α(t ))M −T  = diag(i(α(t )))i(α(t )), V T ATf V −T WT = XT ,  T M −1Ni̇ (α(t ))M −T  = ̇ i(α(t )), BTf V −T WT = Y T ,

Denoting MT Ni−1(α(t ))M = Pi(α(t )), we have −1 Pi̇ (α(t )) = − MT Ni̇ (α(t ))M = − MT Ni−T (α(t ))Ni̇ (α(t ))Ni−1(α(t ))M.

Cf V = Z ,

By replacing MT Ni−1(α(t ))M and −MT Ni−T (α(t ))Ni̇ (α(t ))Ni−1(α(t ))M in (20) by P (α(t )) and P ̇ (α(t )), respectively, and then using Schur

then (26) is identical to (18). It follows from the above discussion and Theorem 1 that the error dynamic (8) is stochastically stable with an H∞ disturbance attenuation level γ. This completes the proof. □

i

i

complement, it is easy to show that

ΦiT (α(t ))Υi (α(t ), α̇(t ))Φi(α(t )) < 0

(21)

which is equivalent to (10). On the other hand, inequality (18) implies that R and W are nonsingular, thus there always exist nonsingular matrices U and V satisfying equality UV ¼W. We define M and  as follows:

Remark 1. The conditions in Theorem 2 involve much valuable information, such as time-varying characteristic of transition rates, the variation rate of parameters and parameter-dependent Lyapunov matrices. In addition, the Lyapunov matrices depend not only on the jumping parameter r(t) but also on the time-varying

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parameter α(t ), which will reduce the conservativeness of the previous results. It should be pointed out that the condition in Theorem 2 cannot be directly checked by LMI Toolbox of Matlab since the parameter-dependent matrix inequality must be hold for all admissible values of the parameter vector α(t ). In the next subsection, we will convert the parameter-dependent matrix inequality into a set of parameter-free LMIs which can be easily solved by using the Matlab LMI toolbox.

to rewrite inequality (18) as M

M−1

Ψi

(l, s)

+ Ψi

(s, l)

< 0, 1 ≤ l ≤ s ≤ M

(27)

where

⎡ ⎢ ⎢ ⎢ (l, s) Ψi = ⎢ ⎢ ⎢ ⎢ ⎣

+  T  iT −  i(l)T

⎡ ⎢ ⎢ ⎢ Ψi(s, l) = ⎢ ⎢ ⎢ ⎢ ⎣

+  T  iT −  i(s)T

*

 i(l, s)

*

*

*

*

*

*

*

 i(s, l)

*

*

*

*

*

 i(l) = ⎡⎣ πi(1l)I

⎤ ⎥ ⎥ i 0 0 ⎥ ⎥, −γ 2I 0 0 ⎥ 0 −I 0 ⎥ (s ) ⎥ * * −diag( i(l))˜ i ⎦

⋯

πi((li)− 1)I

 T  i(l)

l= 1

l= 1 s= l+ 1

∑

(

)

± (υn)  i(n) −  i(M ) ,

(s ) ˜ i = diag  1(s)  2(s) ⋯  i(s−)1  i(s+)1 ⋯  σ(s) ,

(

)

 i(s) = ⎡⎣ πi1(s)I

πi2(s)I

⋯

πi((si −) 1)I

πi((si +) 1)I

⋯

(29)

where

⎡ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

(l, s) Ψ˜i

+  T  iT −  i(l)T *

(l, s) ˜ i

*

*

*

*

*

*

⎤ ⎥ ⎥ i 0 0 ⎥ ⎥, −γ 2I 0 0 ⎥ ⎥ 0 −I 0 (s ) ⎥ * * −diag( i(l))˜ i ⎦ 0

iT

 T  i(l)

with (l, s) ˜ i = πii(l) i(s) −

M−1

∑

(

)

αṅ (t )  i(n) −  i(M ) .

n= 1 M−1

(

)

Note that there exists a term ∑n = 1 αṅ (t )  (in) −  (iM ) in (29). In this case, if all the possible combinations of the upper and lower bounds of (l , s ) (s , l ) (l , l ) α̇n(t ) (n = 1, 2, … , M − 1) ensure that Ψ˜i < 0 and Ψ˜i + Ψ˜i < 0, then matrix inequality (29) holds. It immediately follows from (27) that matrix inequality (18) holds. This completes the proof. □

Remark 3. In order to check the stability condition of Theorem 2 by using the Matlab LMI toolbox, we have converted the parameter-dependent matrix inequality (18) into a set of LMIs (27). It should be pointed out that, however, the set of LMIs (27) are sufficient conditions for the matrix inequality (18). Thus, some optimal feasible solution of (18) may be lost during the conversion to a set of LMIs. This shows that there is still room for improvement in the optimization problem of matrix inequality (18). In recent literature [41], a gradient-based optimization algorithm is developed to convert the non-LMI constrained optimization problem to a projected gradient problem, and thus fast convergence of parameters is achieved. Feasibility and optimization problems of matrix inequality (18) may be solved using the gradient-based optimization algorithm, which deserves further investigation.

 T  i(s)

πi((li)+ 1)I ⋯ πiσ(l)I ⎤⎦,

n= 1

)

M−1

M−1

 i(l, s) = πii(l) i(s) −

(

(l, s) (s, l) αl(t )αs(t ) Ψ˜i + Ψ˜i <0

Remark 2. The notation ∑n = 1 ± (·) in Theorem 3 is used to indicate that every possible combination of +(·) and −(·) should be included in the LMIs. Therefore, for the fixed i, l and s, each inequality means 2M − 1 different LMIs which must be checked.

⎤ ⎥ ⎥ i 0 0 ⎥ ⎥, −γ 2I 0 0 ⎥ 0 −I 0 ⎥ (s ) ˜ (l ) ⎥ * * −diag( i ) i ⎦ iT

0

*

πi(2l)I

iT

0

M

∑ αl2(t )Ψ˜i(l, l) + ∑ ∑

3.3. H∞ filter computation Theorem 3. For a given scalar γ > 0, the error dynamic (8) is stochastically stable with an H∞ disturbance attenuation level γ if there exist symmetric positive definite matrices  (il ) ∈ 2n × 2n ,  (is) ∈ 2n × 2n , and matrices of appropriate dimensions R, S, W, X, Y, such that the following matrix inequalities hold for all i ∈ 

5

πiσ(s)I ⎤⎦,

M−1

 i(s, l) = πii(s) i(l) −

∑

(

4. Numerical examples

)

± (υn)  i(n) −  i(M ) ,

n= 1

(l ) ˜ i = diag  1(l)  2(l) ⋯  i(l−) 1  i(l+) 1 ⋯  σ(l)

(

)

and the symbol ±(υn) indicates every possible combination of +υn and −υn , n = 1, 2, … , M − 1. Moreover, for any given nonsingular matrix V, a suitable filter in the form of (8) is given by

Af = VW −1XV −1, Bf = VW −1Y , Cf = ZV −1. M

M−1

Proof. Recalling that ∑l = 1 αl(t ) = 1, we see that ∑l = 1 αl̇ (t )+ α̇M (t ) = 0 . Taking into account the Assumption 2, the bound on

α̇M (t ) can be expressed by

M−1

|αM ̇ (t )| ≤ ∑l = 1 υl . Note that

M

M−1

∑ αl̇ (t ) i(l) = ∑ l= 1

(

)

α̇n(t )  i(n) −  i(M ) .

(28)

n= 1

Thus, the term ̇ i(α(t )) in matrix inequality (18) can be replaced by M−1

(

)

∑n = 1 αṅ (t )  (in) −  (iM ) .

(∑

M α (t ) l=1 l

2

)

M

Taking M−1

into M

account

Example 1. Consider the following numerical example from [39]: ⎡ −2.5 0.5 −0.1⎤ ⎡ −3 1 0 ⎤ ⎥ ⎢ ⎢ ⎥ A1= ⎢ 0.3 −2.5 1 ⎥ , A2 = ⎢ 0.1 −3.5 0.3 ⎥ , B1 = ⎣ −0.1 0.3 −3.8⎦ ⎣ −0.1 1 −2 ⎦ C1= L1=

[ 0.8 [ 0.5

⎡ 1⎤ ⎢ ⎥, B = 2 ⎢ 0⎥ ⎣ 1⎦

⎡ −0.6 ⎤ ⎢ ⎥ ⎢ 0.5 ⎥ , ⎣ 0 ⎦

0.3 0 ], C 2 = [ −0.5 0.2 0.3], D1 = 0.2, D 2 = 0.5, −0.1 1], L 2 = [ 0 1 0.6 ].

The transition rate matrix is assumed to be time-varying in a convex bounded domain Π (t ) = α1(t )Π (1) + α2(t )Π (2) with the following vertices:

M

i(α(t )) = ∑l = 1 αl(t ) (il ), it immediately follows that

̇ i(α(t )) =

In this section, three numerical simulation examples will be provided to illustrate the effectiveness and less conservativeness of the proposed filter design approach.

the

⎡ −0.35 0.35 ⎤ ⎡ −0.65 0.65 ⎤ , Π (2) = ⎢ . Π (1) = ⎢ ⎣ 0.2 −0.2⎥⎦ ⎣ 0.4 −0.4 ⎥⎦

equality

= ∑l = 1 αl2(t ) + 2 ∑l = 1 ∑s = l + 1 αl(t )αs(t ) = 1, it is possible

When the parameter vector α(t ) is non-accessible, the above

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6

Π(t ) can be viewed as an unknown transition rate matrix which belongs to a domain with two vertices Π (1) and Π (2) . In this case, the optimal H∞ disturbance attenuation level γ obtained in [39] is 0.9264. Under the assumption that the parameter vector α(t ) is accessible and α̇1(t ) ∈ [ − 1, 1], the optimal H∞ disturbance attenuation level γ obtained using Theorem 3 is 0.6743, which shows that the model in the present paper is more general and the resulting condition has less conservativeness. Moreover, when selecting V¼I, the corresponding filter gain matrices are obtained as follows:

⎡ −1.9197 ⎢ Af = ⎢ −0.1121 ⎣ 0.0417 Cf = ⎡⎣ −0.2062

2.8535 1.3528 ⎤ ⎥ −3.9515 0.0063 ⎥, 1.8197 −0.9723⎦ −0.2701 −0.5541⎦⎤.

⎡ −0.4898⎤ ⎢ ⎥ Bf = ⎢ 1.0397 ⎥, ⎣ −0.2671⎦

2

1

0

10

Example 2. Consider the MJS (1) with the following parameters:

⎡ −0.9 0.7 ⎤ ⎡ −0.8 0.5 ⎤ A1= ⎢ , A = , B = ⎣ 0.8 −1.0⎥⎦ 2 ⎢⎣ 0.6 −0.8⎥⎦ 1

⎡ 0.5⎤ ⎡ 0.6 ⎤ ⎢⎣ ⎥, B2 = ⎢⎣ ⎥, 0.3⎦ 0.4 ⎦

20

Time (sec)

30

40

50

Fig. 1. One possible system mode's evolution.

T T C1= ⎣⎡ 1.5 0.7⎤⎦ , C2 = ⎡⎣ 0.5 1.0⎤⎦ , D1 = 1, D2 = 4, L1= ⎣⎡ 0.7 0.4 ⎦⎤, L 2 = ⎡⎣ 0.5 0.2⎤⎦.

The time-varying uncertain transition rate matrix is assumed to take the following form:

⎡ −0.4 − 0.1sin(t ) 0.4 + 0.1sin(t ) ⎤ ⎥ Λ(sin(t )) = ⎢ ⎣ 0.55 + 0.15sin(t ) −0.55 − 0.15sin(t )⎦ ⎡ −0.1 0.1 ⎤ ⎡ −0.4 0.4 ⎤ =⎢ + sin(t )⎢ . ⎣ 0.55 −0.55⎥⎦ ⎣ 0.15 −0.15⎥⎦ The above affine parameter-dependent transition rate matrix can be converted into the polytopic form as follows:

Π (t ) =

1 − sin(t ) (1) 1 + sin(t ) (2) Π + Π 2 2    α1(t )

α2(t )

Fig. 2. State trajectory and estimation of x1(t ) .

where

⎡ −0.3 0.3 ⎤ ⎡ −0.5 0.5 ⎤ , Π (2) = ⎢ . Π (1) = ⎢ ⎣ 0.4 −0.4 ⎥⎦ ⎣ 0.7 −0.7⎥⎦

1 1 It is obvious that the variation rate of α1(t ) belongs to ⎡⎣ − 2 2 ⎤⎦. Solving LMIs (27), the optimal H∞ disturbance attenuation level γ is 0.6324 and the corresponding filter gain matrices are

⎡ −0.5719 −0.0082⎤ ⎡ 0.2011⎤ Af = ⎢ , B = , ⎣ 0.5208 −0.9747⎥⎦ f ⎢⎣ 0.1398⎥⎦ Cf = ⎣⎡ −0.4195 −0.4568⎦⎤. For the purpose of the simulation, we assume the initial conditions x(0) = [ − 1.0 1.0]T , x^(0) = [0 0]T and r0 = 2. The external disturbance is taken as ω(t ) = sin

( ). Fig. 1 gives one possible t 5π

system mode's evolution. Fig. 2 shows the trajectory of x1(t ) and its estimation. Fig. 3 shows the trajectory of x2(t ) and its estimation. It is clear from Figs. 2 and 3 that the filter can effectively estimate the true state of the system. In addition, because the filter is forced to estimate the downward signal before completing the estimation of the upward signal, the peak of the signal cannot be estimated accurately. This is the major source of estimation error.

⎧ ẋ(t ) = A x(t ) + B ω(t ) i i ⎪ ⎨ y(t ) = Cix(t ) ⎪ ⎩ z(t ) = L ix(t )

Example 3. Consider a vertical take-off and landing (VTOL) helicopter system. The system dynamics can be written as [43,42]:

T where x(t ) = ⎡⎣ x1T (t ) x2T (t ) x3T (t ) x4T (t )⎤⎦ is the state vector, x1(t ) is the

Fig. 3. State trajectory and estimation of x2(t ).

(30)

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7

horizontal velocity, x2(t ) is the vertical velocity, x3(t ) is the pitch rate, x4(t ) is the pitch angle, and ω(t ) is the wind gust disturbance. The system dynamics depend on the airspeeds, which are characterized by the Markov process taking values in a finite space  = {1, 2}. These two modes correspond, respectively, to 135 and 170 knots. The system matrices A1 and A2 are developed by using the mode-independent state feedback controller in [42]:

⎡ −0.0571 0.1355 0.2431 −0.2334 ⎤ ⎥ ⎢ −0.2564 −0.8905 −0.2232 −0.7582⎥ , A1 = ⎢ ⎢ 0.4402 −0.3427 −2.0006 −2.7377⎥ ⎥⎦ ⎢⎣ 0 0 1.0000 0 ⎡ −0.0571 0.1355 0.2431 −0.2334 ⎤ ⎥ ⎢ −0.3290 −0.5978 0.4251 0.2639 ⎥ . A2 = ⎢ ⎢ 0.4402 −0.2027 −2.0006 −1.6377⎥ ⎥ ⎢⎣ ⎦ 0 0 1.0000 0 T T B1 = ⎡⎣ 0.01 −0.02 0.01 −0.03⎤⎦ , B2 = ⎡⎣ −0.03 0.03 0.01 0.02⎤⎦ ,

Fig. 5. State trajectory and estimation of x(t).

C1 = C2 = ⎡⎣ 1 0 1 0⎤⎦, L1 = L 2 = ⎡⎣ −0.5 −0.6 0.5 0.3⎤⎦. The global transition rate matrix is described by the following polytopic parameter-dependent matrix:

⎡ −1.0278 1.0278 ⎤ ⎡ −1.6026 1.6026 ⎤ Π (α(t )) = α1(t )⎢ + α2(t )⎢ ⎣ 0.4116 −0.4116⎥⎦ ⎣ 0.7505 −0.7505⎥⎦ where α1(t ) + α2(t ) = 1 and the variation rate of α1(t ) belongs to [ − 1, 1]. Solving the LMIs (27), we get the optimal H∞ disturbance attenuation level γ = 0.083, and the filter gains are found as follows:

⎡ −5.6618 0.0543 −5.0902 −0.4340⎤ ⎥ ⎢ 5.2092 −0.7565 5.7362 0.2876 ⎥ , Af = ⎢ ⎢ −1.0556 −0.0750 −3.4403 −1.6996⎥ ⎢⎣ 4.6463 −0.0156 5.3629 0.0334 ⎦⎥ T Bf = ⎡⎣ 5.6447 −5.6942 1.5248 −4.6840⎤⎦ ,

Cf = ⎡⎣ 0.3310 0.3991 −0.3079 −0.2048⎤⎦. Given the initial conditions x(0) = [ − 0.5 0.5 − 0.5 0.5]T and x^(0) = [0 0 0 0]T . Fig. 4 shows one possible system mode's evolution, Fig. 5 gives the state trajectories and their estimations, and Fig. 6 depicts the trajectory of the estimation error. It can be observed from Figs. 5 and 6 that the filter of this paper can effectively estimate the true state of system.

5. Conclusions This paper has studied the problem of H∞ filter design for a

Fig. 6. Estimation error e(t).

class of continuous-time MJSs with time-varying uncertain transition rates. By constructing a parameter-dependent Lyapunov function and fully considering the information about the variation rate of time-varying parameters, the existence of a mode-independent H∞ filter has been given in terms of a parameter-dependent matrix inequality. Furthermore, a sufficient condition for the solvability of the filter gain matrix has been obtained by converting the parameter-dependent matrix inequality into a finite set of LMIs. The feasibility and effectiveness of the proposed method have been illustrated by three numerical simulation examples.

References 2

1

0

5

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15 Time (sec)

20

25

Fig. 4. One possible system mode's evolution.

30

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[27]

[28] [29]

[30] [31]

[32] [33] [34] [35]

[36]

[37]

[38]

[39] [40]

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Please cite this article as: Liu H, et al. New results on H1 filtering for Markov jump systems with uncertain transition rates. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.04.015i