1 Introduction

The problem of stability analysis for Lur’e control systems has been a topic of recurring interest over the past decades due to the fact that a large class of nonlinear systems in engineering can be modeled as this class of systems. A great number of results related to such systems have been reported in the literatures [1–3]. Recently, much attention has been paid to the study of Lur’e singular systems. The control problem for singular systems of such class is more complicated than for regular ones, because the singular systems usually have three types of mode, namely, finite-dynamic mode, impulse mode and non-dynamic mode, while the latter two do not appear in state space. The results related to Lur’e singular systems can be classified into two types: delay-independent cases[4,5] and delay-dependent cases[6–10]. It should be pointed out that the delay-dependent results are less conservative than the delay-independent ones, especially when the time delays are small. Till now, some results have been reported related to singular/regular systems with Markovian switching[11–15]. However, to the best of our knowledge, the delay-dependent stability analysis for Lur’e singular systems with Markovian switching has not been investigated.

On the other hand, the notion of passivity has played an important role in analysis and control design of linear and nonlinear systems. Passivity is part of a broader and more general theory of dissipativity, and it can maintain the internal stability of the system[16,17]. So it is necessary to investigate the passivity of systems. However, the passivity analysis for Lur’e systems has been paid little attention.

Motivated by the aforementioned factors, the problem of delay-dependent stability and passivity for Lur’e singular systems with Markovian switching is considered in this paper. The free-weighting matrices approach is employed to derive the delay-dependent criterion, which guarantees that the system is regular, impulse free, stochastically stable, and robustly passive. Two numerical examples are provided to illustrate the effectiveness of the proposed method.

2 Problem formulation

Consider a class of Lur’e singular systems with Markovian switching and time-varying delay described as

$$\left\{ {\begin{array}{*{20}{c}} {E\dot x(t) = A({r_t})x(t) + {A_d}({r_t})x(t - d(t)) + D({r_t})f(\sigma ({r_t}),{r_t}) + B({r_t})w(t)} \\ {z(t) = {C_z}({r_t})x(t) + {C_{dz}}({r_t})x(t - d(t)) + {B_z}({r_t})w(t)} \\ {\sigma ({r_t}) = C({r_t})x(t)} \\ {x(t) = \phi (t),\;\,\forall t \in [ - \bar d,0]} \end{array} } \right.$$
(1)

where x(t) ∈ Rn is the state vector, z(t) ∈ Rs is the controlled output, w(t) ∈ Rp is the disturbance input which belongs to L2 [0,∞), E ∈ Rn×n may be singular, which is assumed that rank (E) = r ≥ n, ϕ(t) ∈ \({C_{n,\bar d}}\) is a compatible vector valued initial function, A(r t ), A d (r t ), B( r t ), D(rt), C z ( r t), C dz ( r t), B( r t), B z (r t ) and C (r t ) are real constant matrices of appropriate dimensions, and {r t } is a continuous-time Markov process with a right continuous trajectory taking values finite set S = {1, 2, …, N} with transition probabilities as

$$P\,[{r_{t + \Delta t}} = j|{r_t} = i] = \left\{ {\begin{array}{*{20}{l}} {{\pi _{ij}}\Delta + o(\Delta ),\;{\text{if}}\;j \ne i} \\ {1 + {\pi _{ij}}\Delta + o(\Delta ),\;{\text{if}}\;j = i} \end{array}} \right.$$

where \({\lim _{\Delta \to 0}}\frac{{o(\Delta )}}{\Delta } = 0,\,\,\,{\pi _{ij}} > 0,\,\,\,j\,\,\, \ne i\,{\text{and}}\,\,\,{\pi _{ii}} = - \sum\nolimits_{j \ne i} {{\pi _{ij}}{\pi _{ij}}} \) for each i ∈ S, σ (rt) = ( σ rt1 , σ rt2 , … ,σ rtq )T, frt (σi) = (f rt1 (σ i 1), f rt2 (σi2),…, f rtq (σ iq ))T. The time delay d(t) is a time-varying continuous function that satisfies

$$0 \leqslant d(t) \leqslant \bar d,\;\;0 \leqslant \dot d(t) \leqslant \mu \leqslant 1,\;\;\;\;\forall i \in S$$
(2)

where \(\bar d\) and μ are constants.

For notational simplicity, in the sequel, for each possible r t ∈ i, i ∈ S, a matrix M( r t ) will be denoted by Mi. For example, A (r t ) is denoted by A i , A d (r t ) is denoted by A di , etc.

The nonlinear feedback path is formulated as

$$\begin{array}{*{20}{l}} {{f_i}({\sigma _i}) = {{({f_{i1}}({\sigma _{i1}}),{f_{i2}}({\sigma _{i2}}), \cdots ,{f_{im}}({\sigma _{im}}))}^T}} \\ {{\sigma _i} = {{({\sigma _{i1}},{\sigma _{i2}}, \cdots ,{\sigma _{im}})}^T}} \end{array}$$

with each f ij (•) satisfying the finite sector condition:

$$\begin{array}{*{20}{l}} {{f_{ij}}( \cdot ) \in {K_j}[0,{k_j}] = \{ {f_{ij}}( \cdot )|{f_{ij}}(0) = 0,0 < {\sigma _{ij}}{f_{ij}}({\sigma _{ij}}) \leqslant {k_j}\sigma _{ij}^2,{\sigma _{ij}} \ne 0\} } \\ {j = 1,2, \cdots ,m,\;\,\forall i \in S.} \end{array}$$
(3)

The nominal unforced system of (1) can be written as

$$E\dot x(t) = {A_i}x(t) + {A_{di}}x(t - d(t)).$$
(4)

Throughout this paper, we shall use the following concepts and introduce the following useful lemmas.

Definition 1.

  1. 1)

    System (4) is said to be regular and impulse free for any time delay d(t) satisfying (2), if the pairs (E,A i ) and (E, A i + A di ) are regular and impulse free for every i ∈ S.

  2. 2)

    System (4) is said to be stochastically stable for any f ij (•) (j = 1, 2, • • • , q) satisfying (3), if there exists a constant T (r 0 ,φ(•)) such that

    $$E\left[ {\int_0^\infty {||x(t)|{|^2}} dt|{r_0},x(s) = \phi (s),s \in [ - \bar d,0]} \right] \leqslant T({r_0},\phi ( \cdot )).$$
  3. 3)

    System (4) is said to be stochastically admissible, if it is regular, impulse free and stochastically stable.

Definition 2. The Lur’e singular system (1) is said to be robustly passive, if there exists a scalar γ ≥ 0 such that

$$E\left\{ {2\int_0^{{t^ * }} {{w^T}} (s)z(s)ds} \right\} \geqslant - \gamma \int_0^{{t^ * }} {{w^T}} (s)w(s)ds$$
(5)

for all t* < 0 under zero initial conditions, w(t) ∈ L 2 [0, ∞).

Lemma 1 (S-procedure)!18!. Let F i ∈ R n×n, i = 0,1,2,… ,p, the following statement is true: ζ T Foζ < 0 for any ζ ∈ Rn satisfying ζ TF i ζ ≥ 0, if there exist real scalars τ i ≥ 0, i = 0,1, 2, … ,p, such that

$${F_0} - \mathop \Sigma \limits_{i = 1}^p \;{\tau _i}{F_i} > 0.$$

For p = 1, these two statements are equivalent.

3 Main results

In this section, we shall focus on the delay-dependent stability and passivity for system (1).

Theorem 1. For given scalars d, γ ≥ 0 and 0 ≤ μ ≤ 1, then, for any delay d(t) satisfying (2), system (1) is stochastically stable and robustly passive, if there exist symmetric positive-definite matrices Q, Z, Λ i , a scalar ε i and matrices P i , Y u , Y 2i, Y 3 i , Y M , V U , V 2i and V 3i such that for each i ∈ S,

$${E^T}{P_i} = P_i^TE \geqslant 0$$
(6a)
$${\Theta _i} = \left[ {\begin{array}{*{20}{c}} {{\Xi _{i11}}}&{{\Xi _{i12}}}&{{\Xi _{i13}}}&{\bar d{Y_{1i}}}&{{\Xi _{i15}}}&{V_{1i}^T{B_i} - C_{zi}^T} \\ * &{{\Xi _{i22}}}&{{\Xi _{i23}}}&{\bar d{Y_{2i}}}&{V_{2i}^T{D_i} - {V_{3i}}}&{V_{2i}^T{B_i}} \\ * & * &{{\Xi _{i33}}}&{\bar d{Y_{3i}}}&{{\Xi _{i35}}}&{ - C_{dzi}^T} \\ * & * & * &{ - \bar dZ}&{\bar dY_{4i}^T}&0 \\ * & * & * & * &{{\Xi _{i55}}}&{{\Lambda _i}{C_i}{B_i} + V_{3i}^T{B_i}} \\ * & * & * & * & * &{ - \gamma I - B_{zi}^T - {B_{zi}}} \end{array}} \right] < 0$$
(6b)

where

$$\begin{array}{*{20}{c}} {{\Xi _{i11}} = \mathop \Sigma \limits_{j = 1}^N \;{\pi _{ij}}{E^T}{P_j} + V_{1i}^T{A_i} + A_i^T{V_{1i}} + {Y_{1i}}E + {E^T}Y_{1i}^T + Q} \\ {{\Xi _{i12}} = P_i^T - V_{1i}^T + A_i^T{V_{2i}} + {E^T}Y_{2i}^T} \\ {{\Xi _{i13}} = - {Y_{1i}}E + V_{1i}^T{A_{di}} + {E^T}Y_{3i}^T} \\ {{\Xi _{i15}} = A_i^TC_i^T{\Lambda _i} + {E^T}Y_{4i}^T + V_{1i}^T{D_i} + A_i^T{V_{3i}} + {\varepsilon _i}C_i^TK_i^T} \\ {{\Xi _{i22}} = - {V_{2i}} - V_{2i}^T + \bar dZ} \\ {{\Xi _{i23}} = - {Y_{2i}}E + V_{2i}^T{A_{di}}} \\ {{\Xi _{i33}} = - (1 - \mu )Q - {Y_{3i}}E - {E^T}Y_{3i}^T} \\ {{\Xi _{i35}} = A_{di}^TC_i^T{\Lambda _i} - {E^T}Y_{4i}^T + A_{di}^T{V_{3i}}} \\ {{\Xi _{i55}} = {\Lambda _i}{C_i}{D_i} + D_i^TC_i^T{\Lambda _i} + V_{3i}^T{D_i} + D_i^T{V_{3i}} - 2{\varepsilon _i}I.} \end{array} $$

Proof. First, we shall show system (1) is regular and impulse free.

Define

$$\begin{array}{*{20}{l}} {\bar E = \left[ {\begin{array}{*{20}{c}} E&0 \\ 0&0 \end{array}} \right],{{\bar A}_i} = \left[ {\begin{array}{*{20}{c}} 0&I \\ {{A_i}}&{ - I} \end{array}} \right],} \\ {{{\bar A}_{di}} = \left[ {\begin{array}{*{20}{c}} 0&0 \\ {{A_{di}}}&0 \end{array}} \right],\;\;{{\bar P}_i} = \left[ {\begin{array}{*{20}{c}} {{P_i}}&0 \\ {{V_{1i}}}&{{V_{2i}}} \end{array}} \right]} \\ {{{\bar Y}_{1i}} = \left[ {\begin{array}{*{20}{c}} {{Y_{1i}}}&0 \\ {{Y_{2i}}}&0 \end{array}} \right]\;\;{{\bar Y}_{3i}} = \left[ {\begin{array}{*{20}{c}} {{Y_{3i}}}&0 \\ 0&0 \end{array}} \right],} \\ {{{\bar Q}_1} = \left[ {\begin{array}{*{20}{c}} Q&0 \\ 0&{\bar dZ} \end{array}} \right],\;\;{{\bar Q}_2} = \left[ {\begin{array}{*{20}{c}} {(1 - \mu )Q}&0 \\ 0&{\bar dZ} \end{array}} \right].} \end{array}$$

From (6), it is easy to see that

$${\bar E^T}{\bar P_i} = \bar P_i^T\bar E \geqslant 0$$
(7a)
$${\pi _{ii}}{\bar E^T}{\bar P_i} + \bar A_i^T{\bar P_i} + \bar P_i^T{\bar A_i} + {\bar Y_{1i}}\bar E + {\bar E^T}\bar Y_{1i}^T < 0$$
(7b)
$$\left[ {\begin{array}{*{20}{c}} {\bar \Xi }&{\bar P_i^T{{\bar A}_{di}} - {{\bar Y}_{1i}}\bar E + {{\bar E}^T}\bar Y_{3i}^T} \\ * &{ - {{\bar Q}_2} - {{\bar E}^T}\bar Y_{3i}^T - {{\bar Y}_{3i}}\bar E} \end{array}} \right] < 0$$
(7c)

where

$$\bar \Xi = {\pi _{ii}}{\bar E^T}{\bar P_i} + \bar A_i^T{\bar P_i} + \bar P_i^T{\bar A_i} + {\bar Q_1} + {\bar Y_{1i}}\bar E + {\bar E^T}\bar Y_{1i}^T$$

Since rank(E ) = r ≤ n, there exist nonsingular matrices G and H, such that GEH \(\left[ {\begin{array}{*{20}{c}} {{I_r}}&0 \\ 0&0 \end{array}} \right]\) Denote

$$\begin{array}{*{20}{c}} {G{{\bar A}_i}H = \left[ {\begin{array}{*{20}{c}} {{A_{i11}}}&{{A_{i12}}} \\ {{A_{i21}}}&{{A_{i22}}} \end{array}} \right]} \\ {{G^{ - T}}{{\bar P}_i}H = \left[ {\begin{array}{*{20}{c}} {{P_{i11}}}&{{P_{i12}}} \\ {{P_{i21}}}&{{P_{i22}}} \end{array}} \right]} \\ {{H^T}{{\bar Y}_{1i}}{G^{ - 1}} = \left[ {\begin{array}{*{20}{c}} {{Y_{i11}}}&{{Y_{i12}}} \\ {{Y_{i21}}}&{{Y_{i22}}} \end{array}} \right]} \end{array}$$

for every i ∈ S. From (7a), we can deduce that P i 22 = 0 for each i ∈ S. Pre-multiplying and post-multiplying (7b) by H T and H, respectively, we have

$$A_{i22}^T{P_{i22}} + P_{i22}^T{A_{i22}} < 0.$$
(8)

According to the proof of Theorem 1 in [7], we have from (8) that the pair (E, Ai) is regular and impulse free for each i ∈ S. Pre-multiplying and post-multiplying (7c) by [I I] and [I I] T, respectively, we get

$${\pi _{ii}}{\bar E^T}{\bar P_i} + {({\bar A_i} + {\bar A_{di}})^T}{\bar P_i} + \bar P_i^T({\bar A_i} + {\bar A_{di}}) < 0.$$
(9)

Using the above approach, from (7a) and (9), we can get that the pair (E,Ai + A di ) is regular and impulse free for each i ∈ S. From Definition 1, system (1) with u(t) = 0 is regular and impulse free for any delay d(t) satisfying (2).

Next, we shall prove the stochastic stability of the system (1) with w(t) = 0. Define a new process {(x t , r t ),t ≥ 0} by xt(s) = x(t + s), -\(2\bar d\) ≤ s ≤ 0, then {(x t ,r t ),t ≥ 0} is a Markov process with initial state (ϕ(•), r 0). Now, for t ≥ \(\bar d\), choose a stochastic Lyapunov functional candidate as

$$\begin{array}{*{20}{l}} {V({x_t},{r_t},t) = {x^T}(t){E^T}P({r_t})Ex(t) + } \\ {\int_{t - d(t)}^t {{x^T}} (\alpha )Qx(\alpha )d\alpha + } \\ {\int_{ - d}^0 {\int_{t + \beta }^t {{{\dot x}^T}} } (\alpha ){E^T}ZE\dot x(\alpha )d\alpha d\beta + } \\ {2\sum\limits_{j = 1}^m {{\lambda _{ij}}} \int_0^{{\sigma _{ij}}} {{f_{ij}}} ({\sigma _{ij}})d{\sigma _{ij}}.} \end{array}$$
(10)

Let L be the weak infinitesimal generator of the random process{x t , r t }. Then, for each i∈S,we have

$$\begin{array}{*{20}{c}} {LV({x_t},i,t) \leqslant 2{x^T}(t){E^T}{P_i}\dot x(t) + {x^T}(t)\left( {\sum\limits_{j = 1}^N {{\pi _{ij}}{E^T}{P_j}} } \right)x(t) + } \\ {{x^T}(t)Qx(t) - (1 - \mu ){x^T}(t - d(t))Qx(t - d(t)) + } \\ {\bar d{{\dot x}^T}(t){E^T}ZE\dot x(t) - \int_{t - d(t)}^t {{{\dot x}^T}(\alpha ){E^T}ZE\dot x(\alpha )d\alpha + } } \\ {2f_i^T({\sigma _i}){\Lambda _i}{C_i}[{A_i}x(t) + {A_{di}}(t - d(t)) + {D_i}{f_i}({\sigma _i})] + } \\ {2[{x^T}(t){Y_{1i}} + {{(E\dot x(t))}^T}{Y_{2i}} + {x^T}(t - d(t)){Y_{3i}} + f_i^T({\sigma _i}){Y_{4i}}] \times } \\ {\left[ {Ex(t) - Ex(t - d(t)) - \int_{t - d(t)}^t {E\dot x} (\alpha )d\alpha } \right] + } \\ {2[{x^T}(t)V_{1i}^T + {{(E\dot x(t))}^T}V_{2i}^T + f_i^T({\sigma _i})V_{3i}^T] \times } \\ {[ - E\dot x(t) + {A_i}x(t) + {A_{di}}(t - d(t)) + {D_i}{f_i}({\sigma _i})].} \end{array}$$

It follows from Jensen integral inequality[19] that

$$ - \int_{t - d(t)}^t {{{\dot x}^T}} (\alpha ){E^T}ZE\dot x(\alpha )d\alpha \leqslant {\zeta ^T}(t)( - \bar dZ)\zeta (t)$$
(11)

where ζ(t) = - \(\int_{t - d(t)}^t {\frac{1}{d}E\dot x(\alpha )d\alpha } \).

From (11), we have that, for each i ∈ S

$$LV({x_t},i,t) \leqslant {X^T}{\Phi _{i0}}X$$
(12)

where

$$\begin{array}{*{20}{c}} {X = {{[{x^T}(t){{\dot x}^T}{E^T}{x^T}(t - d(t))\zeta (t)f_i^T({\sigma _i})]}^T}} \\ {{\Phi _{i0}} = \left[ {\begin{array}{*{20}{c}} {{\Xi _{i11}}}&{{\Xi _{i12}}}&{{\Xi _{i13}}}&{\bar d{Y_{1i}}}&{{\Xi _{i15}} - {\varepsilon _i}C_i^TK_i^T} \\ *&{{\Xi _{i22}}}&{{\Xi _{i23}}}&{\bar d{Y_{2i}}}&{V_{2i}^T{D_i} - {V_{3i}}} \\ *&*&{{\Xi _{i33}}}&{\bar d{Y_{3i}}}&{A_{di}^TC_i^T{\Lambda _i} - {E^T}Y_{4i}^T + A_{di}^T{V_{3i}}} \\ *&*&*&{ - \bar dZ}&{\bar dY_{4i}^T} \\ *&*&*&*&{{\Xi _{i55}} + 2{\varepsilon _i}I} \end{array}} \right].} \end{array}$$

In addition, from (3), we have

$$f_i^T({\sigma _i})[{K_i}{C_i}x(t) - {f_i}({\sigma _i})] \geqslant 0.$$
(13)

By applying S-procedure[18], we can see that LV(xt,i,t) < 0 for X ≠ 0 is implied by the existence of a scalar ε i for each i ∈ S such that

$${X^T}{\Phi _{i0}}X + 2{\varepsilon _i}f_i^T({\sigma _i})[{K_i}{C_i}x(t) - {f_i}({\sigma _i})] < 0,\;\;\;\;\forall X \ne 0.$$

That is

$$LV({x_t},i,t) \leqslant {X^T}{\Phi _{i1}}X < 0,\;\;\;\;\forall X \ne 0$$
(14)

where

$${\Phi _{i1}} = {\Phi _{i0}} + \left[ {\begin{array}{*{20}{c}} 0&0&0&0&{{\varepsilon _i}C_i^TK_i^T} \\ *&0&0&0&0 \\ *&*&0&0&0 \\ *&*&*&0&0 \\ *&*&*&*&{ - 2\varepsilon I} \end{array}} \right].$$

From (14), there exists a small scalar ρ < 0 such that for each i ∈ S,L V(x t ,i,t) ≤ -ρ \\x(t)\\ 2.

Therefore, for any t ≥ d, by Dynkin’s formula, we obtain

$$EV({x_t},i,t) - EV({x_{\bar d}},{r_{\bar d}},\bar d) \leqslant - \rho E\int_{\bar d}^t {||x(s)|{|^2}ds} $$

which yields

$$E\int_{\bar d}^t {||x(s)|{|^2}ds} \leqslant {\rho ^{ - 1}}EV({x_{\bar d}},{r_{\bar d}},\bar d).$$
(15)

From [18], the regularity and the absence of impulses of pair (E, Ai) for each i ∈ S implies there exist two invertible matrices Mi and Ni such that

$$\begin{array}{*{20}{l}} {{M_i}E{N_i} = \left[ {\begin{array}{*{20}{c}} {{I_r}}&0 \\ 0&0 \end{array}} \right]} \\ {{M_i}{A_i}{N_i} = \left[ {\begin{array}{*{20}{c}} {{A_{i1}}}&0 \\ 0&{{I_{n - r}}} \end{array}} \right]} \\ {{M_i}{A_{di}}{N_i} = \left[ {\begin{array}{*{20}{c}} {{A_{id1}}}&{{A_{id2}}} \\ {{A_{id3}}}&{{A_{id4}}} \end{array}} \right]} \\ {{M_i}{D_i} = \left[ {\begin{array}{*{20}{c}} {{D_{i1}}} \\ {{D_{i2}}} \end{array}} \right]} \\ {{{\bar C}_i} = {C_i}N_i^{ - 1}.} \end{array}$$

Then, for each i ∈ S, system (1) is decomposed as

$$\begin{array}{*{20}{c}} {{{\dot \xi }_1}(t) = {A_{i1}}{\xi _1}(t) + {A_{id1}}{\xi _1}(t - d(t)) + {A_{id2}}{\xi _2}(t - d(t)) + {D_{i1}}{f_i}({\eta _i})} \\ { - {\xi _2}(t) = {A_{id3}}{\xi _1}(t - d(t)) + {A_{id4}}{\xi _2}(t - d(t)) + {D_{i2}}{f_i}({\eta _i})} \\ {\xi (t) = \psi (t) = N_i^{ - 1}\phi (t),t \in [ - \bar d,0]} \end{array} $$
(16)

where \({\eta _i} = {\bar C_i}\xi (t),\xi (t) = \left[ {\begin{array}{*{20}{c}} {{\xi _1}(t)} \\ {{\xi _2}(t)} \end{array}} \right] = N_i^{ - 1}x(t)\)

Expression (3) can be equivalently described as

$$\begin{array}{*{20}{l}} {{\sigma _{ij}}{f_{ij}}({\sigma _{ij}})\;({\sigma _{ij}}{f_{ij}}({\sigma _{ij}}) - {k_j}\sigma _{ij}^2) \leqslant 0\;(j = 1,2, \cdots ,m) \Rightarrow } \\ {f_{ij}^2({\sigma _{ij}}) \leqslant k_j^2\sigma _{ij}^2 \Rightarrow ||{f_i}({\sigma _i})|{|^2}\; \leqslant \;\;||{K_i}{C_i}x(t)|{|^2}.} \end{array}$$
(17)

Therefore,

$$f_i^T{f_i} \leqslant {\xi ^T}(t)\bar C_i^TK_i^T{K_i}{\bar C_i}\xi (t).$$
(18)

From (17), we can deduce that

$$\begin{array}{*{20}{l}} {||{D_{i1}}{f_i}({\eta _i})|{|^2} \leqslant \;||{D_{i1}}{K_i}{{\bar C}_i}\xi (t)|{|^2} \Rightarrow \,||{D_{i1}}{f_i}({\eta _i})||\; \leqslant \;\sqrt {{k_1}} ({\xi _1}(t) + {\xi _2}(t))} \\ {||{D_{i1}}{f_i}({\eta _i})|{|^2} \leqslant \;||{D_{i2}}{K_i}{{\bar C}_i}\xi (t)|{|^2} \Rightarrow \,||{D_{i2}}{f_i}({\eta _i})||\; \leqslant \;\sqrt {{k_2}} ({\xi _1}(t) + {\xi _2}(t)).} \end{array}$$

where

$$\begin{array}{*{20}{c}} {{k_1} = {\lambda _{\max ,i \in \,S}}\left[ {\bar C_i^TK_i^TD_{i1}^T{D_{i1}}{K_i}{{\bar C}_i}} \right] \geqslant 0} \\ {{k_2} = {\lambda _{\max ,i \in \,S}}\left[ {\bar C_i^TK_i^TD_{i2}^T{D_{i2}}{K_i}{{\bar C}_i}} \right] \geqslant 0.} \end{array} $$

For any 0 ≤ t ≤ \(\bar d\), it follows from (16) that

$$||{\xi _1}(t)||\; \leqslant \;||{\xi _1}(0)|| + {k_3}\int_0^t {[||{\xi _1}(s)|| + ||{\xi _1}(s - d(t))|| + ||{\xi _2}(s)|| + ||{\xi _2}(s - d(t))||]} \,ds \leqslant (2{k_3}\bar d + 1)||\psi |{|_{\bar d}} + {k_3}\int_0^t {||{\xi _1}(s)||ds} $$

where k 3 = max i∈S \(\left\{ {||{A_{i1}}|| + \sqrt {{k_1}} ,||{A_{id1}}||,||{A_{id2}}||} \right\} \geqslant 0\).

By applying the Gronwall-Bellman Lemma, we obtain from (18) that for any 0 ≤ t ≤ \(\bar d\),

$$\mathop {\sup }\limits_{0 \leqslant t \leqslant \bar d} ||{\xi _1}(t)||\; \leqslant \;(2{k_3}\bar d + 1)||\psi |{|_{\bar d}}{e^{{k_3}\bar d}}.$$
(19)

Consider (16) and (19), it can be deduced that sup0≤t≤d ||ξ 2( t )|| is bounded. We assume \(0 \leqslant \sqrt {{k_2}} < 1\), thenfor any 0 ≤ t ≤ \(\bar d\), we have

$$\mathop {\sup }\limits_{0 \leqslant t \leqslant \bar d} ||{\xi _2}(t)|{|^2}\; \leqslant {\left( {1 - \sqrt {{k_2}} } \right)^{ - 2}}k_4^2\;{\left( {(2{k_3}\bar d + 1){e^{{k_3}\bar d}} + 2} \right)^2}||\psi |{|_{\bar d}}$$

where k 4 = max i∈ S \(\left\{ {||{A_{id3}}||,||{A_{id4}}||,\sqrt {{k_2}} } \right\}\). Hence,

$$\mathop {\sup }\limits_{0 \leqslant t \leqslant \bar d} ||\xi (t)||\; \leqslant \mathop {\sup }\limits_{0 \leqslant t \leqslant \bar d} ||{\xi _1}(t)|{|^2} + \mathop {\sup }\limits_{0 \leqslant t \leqslant \bar d} ||{\xi _2}(t)|{|^2} \leqslant {k_5}||\psi ||_{\bar d}^2$$

where

$${k_5} = {(2{k_3}\bar d + 1)^2}{e^{2{k_3}\bar d}} + {\left( {1 - \sqrt {{k_2}} } \right)^{ - 2}}k_4^2 \times {\left( {(2{k_3}\bar d + 1){e^{{k_3}\bar d}} + 2} \right)^2}.$$

Therefore,

$$\mathop {\sup }\limits_{0 \leqslant t \leqslant \bar d} ||x(t)|{|^2} \leqslant {k_5}||{N_i}|{|^2}||N_i^{ - 1}|{|^2}||\phi ||_{\bar d}^2.$$
(20)

Expressions (15) and (20) imply the existence of a scalar k 6 ≥ 0 such that

$$E\int_0^t {||x(s)|{|^2}} ds = E\int_0^{\bar d} {||x(s)|{|^2}} ds + E\int_{\bar d}^t {||x(s)|{|^2}} ds \leqslant {k_6}E||\phi ||_{\bar d}^2.$$

From Definition 1, we get that system (1) with w(t) = 0 is stochastically stable.

Now we shall show the passivity property of system (1) for each i ∈ S.

Under zero initial condition, it is easy to see that

$$\begin{array}{*{20}{l}} {LV({x_t},i,t) - 2{w^T}(t)z(t) - \gamma {w^T}(t)w(t) = } \\ {LV({x_t},i,t) - \gamma {w^T}(t)w(t) - } \\ {2{w^T}(t)[{C_z}({r_t})x(t) + {C_{dz}}({r_t})x(t - d(t)) + {B_z}({r_t})w(t)] \leqslant } \\ {{{\bar X}^T}(s){\Theta _i}\bar X(s)} \end{array}$$

where

$$\bar X = {\left[ {{x^T}(t){{\dot x}^T}{E^T}{x^T}(t - d(t)){\zeta ^T}(t)f_i^T({\sigma _i}){w^T}(t)} \right]^T}.$$

From (6b), we get

$$LV({x_t},i,t) - 2{w^T}(t)z(t) - \gamma {w^T}(t)w(t) < 0.$$
(21)

Integrating both sides of (21) with respect to t over the time period [0, t*] gives rise to

$$V({x_{t*}},i,{t^ * }) - V({x_0},i,0) - E\left\{ {s\int_0^{{t^ * }} {{w^T}} (s)z(s)ds} \right\} - \gamma \int_0^{{t^ * }} {{w^T}} (s)w(s)ds \leqslant 0.$$

Then, from Definition 2, system (1) is robustly passive. □

Remark 1. Theorem 1 gives a delay-dependent sufficient condition of the stochastic stability with passivity for Lur’e singular system with Markovian switching (1). In the derivation of the delay-dependent result in Theorem 1, no model transformation is performed onto system (1). Moreover, we have introduced some free-weighting matrices, (this approach is also known as the slack matrix approach), which may help to reduce conservatism.

4 Numerical examples

In this section, two numerical examples are presented to illustrate the effectiveness of the proposed method.

Example 1. Consider system (1) with one mode and the system parameters are described as

$$\begin{array}{*{20}{l}} {{A_1} = \left[ {\begin{array}{*{20}{c}} { - 0.5}&0 \\ 0&{ - 1} \end{array}} \right],\;\;{A_{d1}} = \left[ {\begin{array}{*{20}{c}} { - 1.1}&1 \\ 0&{0.5} \end{array}} \right],} \\ {{D_1} = \left[ {\begin{array}{*{20}{c}} {0.2} \\ {0.1} \end{array}} \right],\;\;{B_1} = \left[ {\begin{array}{*{20}{c}} {0.1} \\ {0.2} \end{array}} \right],} \\ {{C_1} = [0.6\;0.8],\;\;{C_{dz1}} = [0.6\;0.8],} \\ {{C_{dz1}} = [0.1\;0.1],\;\;{B_{z1}} = 0.3,} \\ {{K_1} = 0.4.} \end{array}$$

When w(t) is not considered, the maximum allowable delay bound \(\bar d\) for different μ is shown in Table 1. For comparison, the upper bounds obtained for the criteria in [9] are also listed in Table 1. It can be seen that our method is less conservative than those in [9].

Table 1 Maximum allowable time delay \(\bar d\) for different μ
Full size table

Taking disturbance w(t) into account, Table 2 provides the maximum allowed time-delay \(\bar d\) for different γ ≥ 0 when μ = 0.5 according to Theorem 1.

Table 2 Maximum allowable time delay d for different γ
Full size table

Example 2. Consider the system (1) with two modes, i.e., N = 2, the system parameters are described as

$$\begin{array}{*{20}{l}} {{A_1} = \left[ {\begin{array}{*{20}{c}} { - 0.3}&{0.1} \\ {0.1}&{ - 1} \end{array}} \right],\;\;{A_2} = \left[ {\begin{array}{*{20}{c}} {0.4}&{0.2} \\ 0&{ - 0.9} \end{array}} \right],} \\ {{A_{d1}} = \left[ {\begin{array}{*{20}{c}} { - 1.4}&{0.5} \\ {0.1}&{0.7} \end{array}} \right],\;\;{A_{d2}} = \left[ {\begin{array}{*{20}{c}} { - 1.3}&{0.6} \\ {0.0}&{0.9} \end{array}} \right],} \\ {{B_1} = \left[ {\begin{array}{*{20}{c}} {0.01} \\ {0.02} \end{array}} \right],\;\;{B_2} = \left[ {\begin{array}{*{20}{c}} {0.01} \\ {0.02} \end{array}} \right],} \\ {{D_1} = {D_2} = 0.} \end{array}$$

The rate matrix Π and singular matrix E is chosen as follows:

$$\begin{array}{*{20}{l}} {\Pi = \left[ {\begin{array}{*{20}{c}} { - 0.5}&{0.5} \\ {0.3}&{ - 0.3} \end{array}} \right]} \\ {E = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right].} \end{array}$$

The comparison of maximum time delay upper bound \(\bar d\) via different method is recorded in Table 3.

Table 3 Maximum allowable time delay \(\bar d\) for different μ
Full size table

It can be seen from Table 3 that our results have less conservatism than the recent results for time-delay singular Markovian jump systems in [13].

5 Conclusions

The problem of stochastic stability and passivity for a class of Lur’e singular systems with time-varying delay and Markovian switching has been investigated. The delay-dependent condition has been derived in terms of LMI, in which some free-weighting matrices have been employed to express the relationship between the terms in the Leibniz-Newton formula. Finally, numerical examples have been presented to show the effectiveness and superiority of the proposed method.