A Novel Feedback Control Approach for Networked Systems with Probabilistic Delays

Abstract

The networked control system (NCS) design for continuous-time systems with probabilistic delays is discussed in this paper. The delay is assumed to follow a given probability density function. A novel design scheme for the output feedback controller is developed to render the closed-loop networked system exponentially mean-square stable with $H_{\infty }$ performance requirement. A numerical example is provided to show the advantages of the proposed technique.

1 Introduction

In many modern complex and distributed control systems, systems with remotely located sensors, actuators, controllers and filters are often connected over a sharing communication network. Such architectures are often called networked-control systems (NCS), which bring a lot of advantages such as low cost, simple installation and maintenance, increased system agility and so on [1]. The sharing network however makes the analysis and synthesis of such network-based systems challenging. Recently, NCS has attracted much research interest [2]. So far, there has been considerable research work appeared to address modelling, stability analysis, control and filtering problems for NCSs, [3]. Most of the studies on NCSs have concentrated on state feedbacks [4], and the commonly investigated systems have been discrete-time models, sampled-data models, continuous-time models through sampled-data feedback controls. Upon unavailable state information, observer-based feedbacks have to be performed to achieve prescribed control purposes [5–13].

As has been mentioned above that it is difficult to deal with the NCS with long time-varying or random delays, and one aspect of the difficulties lies in providing an appropriate modeling method for such NCSs. Since the delay may be larger than one sampling period, more than one control signals may arrive at the actuator during one sampling interval. Moreover, the numbers of the arriving control signals vary over different sampling intervals, thus the dynamic model of the overall closed-loop NCS varies from sampling period to sampling period [14]. So, the closed-loop NCS is naturally a switched system with the subsystems describing various system dynamics on the different sampling intervals. The switched system model has been used to describe the NCS with delays [15]. However, it is assumed in most of the existing results that the delay is smaller than one sampling period. In [16], the switched system model was used to describe the NCS with long time-varying delays. However, the arbitrary switching scheme was used, which may be conservative and infeasible when some subsystems of the NCS are unstable. Recently, the observer-based feedback controls have been further studied for discrete-time NCSs with random measurements and time delays. In [17], the closed-loop system was transformed into a delay-free model, and an observer-based $\fancyscript{H}_{\infty }$ control design scheme was given in terms of a linear matrix inequality (LMI) to render the closed-loop systems exponentially mean-square stable.

Motivated by the above observations, in this paper, we provide a generalized approach to treating NCSs with probabilistic delays. Specifically, we build on [18] and extend it further to study the problem of the exponential stability of NCSs with probabilistic time-varying delay. By adopting a Lyapunov–Krasovskii functional (LKF) approach and linear matrix inequalities (LMIs), new criteria for the exponential stability of such NCSs are derived in the form of feasibility testing of LMIs, which can be readily solved by using standard numerical software based on inner-minimization methods. We also adopt an appropriate free-weighting matrix method [19] suitable for the derivation of the main results for our considered problem. Numerical example is provided to illustrate that when the variation probability of the time delay is given, the upper bound of the time delay could be much larger than that when only the variation range of the time delay is known.

Notation: We use I and 0 to denote, respectively, the identity matrix and the zero matrix with compatible dimensions; the superscripts T and ‘−1’ stand for the matrix transpose and inverse, respectively; W > 0 means that W is a real symmetric positive definite matrix; ${\parallel } \cdot {\parallel }$ is the spectral norm; ${\mathbf{E}}{\kern 1pt} \left\{ \cdot \right\}$ denotes the expectation and ${\mathbf{Pr}}\left\{ \cdot \right\}$ means the probability; $\lambda_{max} ( \cdot )$ and $\lambda_{min} ( \cdot )$ denote, respectively, the maximum eigenvalue and the minimum eigenvalue of a matrix. In symmetric block matrices, we use the symbol $\bullet$ to represent a term that is induced by symmetry.

2 Problem Formulation

Consider a continuous-time system described by

$$ \begin{aligned} \dot{x}(t) & =\,Ax(t) + Bu(t) + B_{xw} w(t), \\ z(t) & =\,A_{z} x(t) + B_{z} u(t) + B_{zw} w(t), \\ y(t) & =\,Cx(t) + C_{z} u(t) + B_{yw} w(t) \\ \end{aligned} $$

(1)

where $x(t) \in R^{n} ,\;u(t) \in R^{m} ,\;z(t) \in R^{p}$, and $w(t) \in R^{q}$ are the state, the control input, the controlled output and the disturbance input belonging to $\fancyscript{L}_{2} [0,\infty )$, respectively. $A,B,B_{xw} ,B_{z} ,B_{zw} ,C,C_{z} ,B_{yw}$ and $C_{z}$ are known constant real matrices with appropriate dimensions. The pair (A, B) is assumed stabilizable. The measured output $y(t) \in R^{r}$ frequently experiences sensor delay, and it can be described by two random events:

$$ \left\{ {\begin{array}{*{20}l} {Event\,1:\;y(t)} \hfill & {dose\;not\;experience\;sensor\;delay,} \hfill \\ {Event\,2:\;y(t)} \hfill & {experience\;sensor\;delay,} \hfill \\ \end{array} } \right. $$

Recall from the theory of functional differential equations that a continuous and piecewise differentiable initial condition guarantees the existence of the solutions. Assume that the measurement delay $\tau (t)$ from sensor to controller is a random variable whose density function is given by $p(\tau ;\,\pi (t))$, where $\pi (t)$ is a vector of parameters of p. In this paper, we assume that the experience sensor delay distribution is stationary, that is, $\pi (t) = \pi$, where π is a given vector. For example, if p is the normal density function, then $\pi (t) = \left\{ {\mu (t),\,\sigma (t)} \right\}$, where μ(t) and σ(t) are the mean and variance of τ(t). If the support of p contains values that the experience sensor delay cannot attain such as negative values, one could truncate the density function p to have a specified range $\left[ {0,\vartheta } \right]$. In this case, the truncated distribution, p _T is given by

$$ f_{T} \left( {\tau ;\pi (t)} \right) = \frac{{f\left( {\tau ;\pi (t)} \right)}}{{\int_{\alpha }^{\beta } {f\left( {r;\pi (t)} \right)dr} }},\quad \rho_{1} \le \tau (t) \le \rho_{2} $$

(2)

Next, consider partitioning the range [α, β] into n mutually exclusive partitions whose end points are: $\left[ {\tau_{0} ,\tau_{1} } \right]\left[ {\tau_{1} ,\tau_{2} } \right] \ldots \left[ {\tau_{n - 2} ,\tau_{n - 1} } \right]\left[ {\tau_{n - 1} ,\tau_{n} } \right]$ where $\tau_{0} = \rho_{1} ,\tau_{n} = \rho_{2}$. Let $\rho_{j} = {\mathbf{Pr}}\left( {\tau_{j - 1} \le \tau (t) \le \tau_{j} } \right)$. Define the indicator functions $\varphi_{j} (t)$ as follows

$$ \varphi_{j} (t) = \left\{ \begin{aligned}& 1:\quad \tau_{j - 1} \le \tau (t) \le \tau_{j} , \hfill \\& 0:\quad otherwise, \hfill \\ \end{aligned} \right. $$

(3)

Further we introduce the time-varying sensor delay $\tau_{j} (t),\;j = 1, \ldots ,n$ where $\tau_{j - 1} \le \tau_{j} (t) \le \tau_{j}$. We will consider the application where the sensor delay τ(t) is stationary, that is, $\mu (t) = \mu$ and $\upsigma (t) =\upsigma$, for all t. Observe that

$$ \begin{aligned} {\mathbf{Pr}}(\varphi_{j} = 1) & =\, {\mathbf{Pr}}(\tau_{j - 1} \le \tau (t) \le \tau_{j} ) = \rho_{j} , \\ {\mathbf{Pr}}(\varphi_{j} = 0) & = 1 - \rho_{j} \\ \end{aligned} $$

(4)

$$ {\kern 1pt} {\mathbf{E}}(\varphi_{j} ) = \rho_{j} ,\;\;{\mathbf{Var}}(\varphi_{j} ) = \rho_{j} (1 - \rho_{j} ) $$

(5)

In this paper, we consider that the time–delay τ(t) satisfies

$$ \rho_{1} \le \tau (t) \le \rho_{2} ,\quad \dot{\tau }(t) \le h,\quad 0 \le \rho_{1} < \rho_{2} $$

(6)

Let the full-order dynamic observer-based feedback control be

$$ \dot{\hat{x}}(t) = K_{a} \hat{x}(t) + K_{c} y(t),\quad \quad u(t) = K_{b} \hat{x}(t), $$

(7)

where $\hat{x} \in R^{n}$ is the observer state, and the feedback gains K _a , K _b and K _c are to be designed. Denote $\delta (t) = \left[ {x(t)^{T} \hat{x}(t)^{T} } \right]^{T}$ and $\rho = {\text{diag}}\left\{ {\rho_{1} , \ldots ,\rho_{n} } \right\}$. Then the closed-loop system of (1) with (4) and (7) is described by

$$ \begin{aligned} \dot{\delta }(t) & = M\delta (t) + M_{\tau } \delta (t - \tau (t)) + B_{\delta w} w(t) \\ & \quad+ \sum\limits_{j = 1}^{n} {\left( {\varphi (t) - \rho_{j} } \right)\left[ {N\delta (t) + N_{\tau } \delta \left( {t - \tau (t)} \right)} \right]} \\ z(t) &= M_{z} \delta (t) + B_{zw} w(t), \end{aligned} $$

(8)

where

$$ \begin{aligned} M & = \left[ {\begin{array}{*{20}c} A & {BK_{b} } \\ {\rho K_{c} C} & {K_{a} } \\ \end{array} } \right], \quad N = \left[ {\begin{array}{*{20}c} 0 & 0 \\ {K_{c} C} & 0 \\ \end{array}} \right], \quad B_{\delta w} = \left[ {\begin{array}{*{20}c} {B_{xw} } \\ {K_{c} B_{yw} } \\ \end{array} } \right], \\ M_{\tau }& = \left[ {\begin{array}{*{20}c} 0 & 0 \\ {(I - \rho )K_{c} D} & 0 \\ \end{array} } \right], \quad N_{\tau } = \left[ {\begin{array}{*{20}c} 0 & 0 \\ { - K_{c} D} & 0 \\ \end{array} } \right], \quad M_{z} = \left[ {\begin{array}{*{20}c} {C_{z} } & {B_{z} K_{a} } \\ \end{array} } \right] \\ \end{aligned} $$

(9)

Here, although the dynamic of the closed-loop system requires only initial values of $\hat{x}(0),\;w(0)$ and $x(t) = \phi (t)\left( {t \in [ - \rho_{2} ,0]} \right)$, for later convenience, we extend the range of the definition of ϕ(t) from $[ - \rho_{2} ,0]$ to $[ - 2\rho_{2} ,0]$ and define a continuous function $\hat{\phi }(t)$ on $[ - 2\rho_{2} ,0]$ such that $\hat{\phi }(t) = \hat{x}(t)$. So, we have $\xi = \left[ {\phi (t)^{T} \hat{\phi }(t)^{T} } \right]^{T}$ for $t \in \left[ { - \rho_{2} ,0} \right]$. We also define w(t) = 0 for $t \in [ - \tau_{0} ,0)$. In the sequel, we let

$$ \begin{aligned} f(\delta ,t): &= M\delta (t) + M_{\tau } \delta (t - \tau (t)) + B_{\delta w} w(t), \\ g(\delta ,t): &= N\delta (t) + N_{\tau } \delta (t - \tau (t)). \\ \end{aligned} $$

(10)

Since f(δ, t) and g(δ, t) in (8) satisfy the local Lipschitz condition and the linear growth condition, the existence and uniqueness of solution to (8) is guaranteed [19]. Moreover, under v(t) = 0 for $t \in [ - \tau_{0} ,0)$, it admits a trivial solution (equilibrium) $\delta \equiv 0$. In this work we will follow the definitions of stochastic stability and $H_{\infty }$ performance requirements.

Definition 1

System (8) is said to be exponentially mean-square stable (EMS) if there exist constants a > 0 and b > 0 such that

$$ {\mathbf{E}}{\kern 1pt} \left\{ {\left\| {\delta (t)} \right\|^{2} } \right\} \le ae^{ - bt} \mathop {\sup }\limits_{{\sigma \, \in \,[ - 2\rho_{2} ,0]}} \;{\mathbf{E}}\left\{ {\left\| {\delta (\sigma )} \right\|^{2} } \right\} $$

(11)

Definition 2

Given η > 0, system (8) is said to be EMS with $H_{\infty }$ performance (EMS-η) if under zero-initial conditions, it is EMS and satisfies

$$ \int\limits_{0}^{\infty } {{\mathbf{E}}\left\{ {\left\| {z(t)} \right\|^{2} } \right\}dt} {\kern 1pt} \; \le \;\eta^{2} \int\limits_{0}^{\infty } {{\mathbf{E}}\left\{ {\left\| {w(t)} \right\|^{2} } \right\}dt} $$

(12)

Controller for system (8) to be EMS-η will be designed.

3 Main Results

Due to the special structure of matrices M _τ and N _τ in system (8), one may choose $\left[ {I_{n} \;0} \right]\delta = x$ to construct certain terms of Lyapunov functionals in order to establish stability conditions [20]. In this work, the full information of δ is used to construct a suitable functional $J(\delta_{t} ,t)$ and a similar type Lyapunov functional $V(\delta_{t} ,t)$ in our study. In details, motivated by recent construction type for retarded systems in [20], we chose the following type of functionals suitable for system (8) to investigate the $H_{\infty }$ performance analysis:

$$ J(\delta_{t} ,t) = J_{1} (\delta_{t} ,t) + J_{2} (\delta_{t} ,t) + J_{3} (\delta_{t} ,t) $$

(13)

where $\delta_{t} = \delta (t + \sigma ),\tau \in [ - 2\rho_{2} ,0]$ and

$$ J_{1} (\delta_{t} ,t) = \delta^{T} (t)P\delta (t),\;J_{2} (\delta_{t} ,t) = \int\limits_{t - \tau (t)}^{t} {\delta^{T} (s)Q\delta (s)ds} + \sum\limits_{i = 1}^{2} \int\limits_{{t - \tau_{i} }}^{t} {\delta^{T} (s)Q_{i} \delta (s)ds} $$

$$ \begin{aligned} J_{3} (\delta_{t} ,t) = & \int\limits_{{ - \rho_{2} }}^{0} {\int\limits_{t + \theta }^{t} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} Z\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]dsd\theta } } \\ & \quad + \int\limits_{{ - \rho_{2} }}^{{ - \rho_{1} }} {\int\limits_{t + \theta }^{t} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} Z_{1} \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]dsd\theta } } \\ \end{aligned} $$

(14)

in which $\hat{\rho }_{j} = \sqrt {\rho_{j} (1 - \rho_{j} )} ,\;j = 0, \ldots ,n,\;\,\varphi_{0} = {\text{diag}}\left\{ {\hat{\rho }_{1} , \ldots ,\hat{\rho }_{n} ,0_{n} } \right\}$, and $P > 0,Q > 0,Q_{1} > 0,Q_{2} > 0,Z > 0$ and $Z_{1} > 0$ are to be determined. For system (8) with w(t) = 0, we use the following Lyapunov functional to obtain EMS conditions:

$$ V(\delta_{t} ,t) = V_{1} (\delta_{t} ,t) + V_{2} (\delta_{t} ,t) + V_{3} (\delta_{t} ,t), $$

(15)

where $V_{i} (\delta_{t} ,t) = J_{i} (\delta_{t} ,t)$ with w(t) = 0, i = 1, 2, 3. Moreover, we use ${\fancyscript{L}} V$ to denote the infinitesimal operator of V [20], which is defined by

$$ \fancyscript{L} V(\delta_{t} ,t) = \mathop {\lim }\limits_{{\varDelta \to 0^{ + } }} \frac{1}{\varDelta }[{\mathbf E} \{ V((\delta_{t + \varDelta } ,t + \varDelta )|(\delta_{t} ,t))\} \; - V(\delta_{t} ,t )] $$

(16)

The following lemma shows that certain condition could ensure system (8) to be EMS.

Lemma 1

Suppose that $K_{a} ,K_{b} ,K_{c} ,P > 0,Q > 0,Q_{i} > 0,Z > 0$ and $Z_{1} > 0$ are given, and $V(\varphi_{t} ,\,t)$ is chosen as in (15). If there exists a constant c > 0 such that

$$ \varvec{E}{\kern 1pt} \{ \fancyscript{L} V(\delta_{t} ,t)\} \le - c{\kern 1pt} \varvec{E}\{ \delta (T)\} $$

holds for all $t \ge 0$ , then system (8) is EMS.

Proof

By Definition 1, the proof is similar to [19].□

The next lemma will be used to establish the analytical result for EMS-η.

Lemma 2

Let $\varSigma ,\varSigma_{1} \; \in \;R^{p \times p}$ be symmetric constant matrices. Then,

$$ \varSigma + \tau (t)\varSigma_{1} < 0 $$

holds for all $\tau (t)\; \in \;[\rho_{1} ,\rho_{2} ]$ if and only if the following two inequalities hold:

$$ \varSigma + \rho_{1} \varSigma_{1} < 0,\;\;\;\varSigma + \rho_{2} \varSigma_{1} < 0 $$

If this is the case, for any $z(t) \in R^{p}$ , the following is true

$$ \quad z(t)^{T} \left( {\varSigma + \tau (t)\varSigma_{1} } \right)z(t) \le \hbox{max} \left\{ {\lambda_{max} (\varSigma + \rho_{1} \varSigma_{1} ),\lambda_{max} (\varSigma + \rho_{2} \varSigma_{1} )} \right\}{\parallel }z(t){\parallel }^{2} $$

Proof

For any $\tau (t) \in [\rho_{1} ,\rho_{2} ]$, there exists an $\alpha_{t} \; \in \;[0,\,1]$ such that $\tau (t) = \alpha_{t} \rho_{1} + (1 - \alpha_{t} )\rho_{2}$. This gives $\varSigma + \tau (t)\varSigma_{1} = \alpha_{t} (\varSigma + \rho_{1} \varSigma_{1} ) + (1 - \alpha_{t} )(\varSigma + \rho_{2} \varSigma_{1} ) < 0$. Then

$$ \begin{aligned} & z(t)^{T} (\varSigma + \tau (t)\varSigma_{1} )z(t) \le \alpha_{t} \lambda_{max} (\varSigma + \rho_{1} \varSigma_{1} ){\parallel }z(t){\parallel }^{2} + \lambda_{max} (\varSigma + \rho_{2} \varSigma_{1} ){\parallel }z(t){\parallel }^{2} \\ & \le \hbox{max} \left\{ {\lambda_{max} (\varSigma + \rho_{1} \varSigma_{1} ),\lambda_{max} (\varSigma + \rho_{2} \varSigma_{1} )} \right\}{\parallel }z(t){\parallel }^{2} \\ \end{aligned} $$

Theorem 1

Given η > 0, the closed-loop system (8) is EMS-η if there exist 2n × 2n matrices $P > 0,\;Q > 0,\;Q_{1} > 0$ and $Q_{2} > 0,\,4n \times 4n$ matrices $Z > 0,Z_{1} > 0,L_{1} > 0,L_{2} > 0$ and $L_{3} > 0,\;(8n + q)\; \times \;2n$ matrices F, G and H, such that

$$ \left[ {\begin{array}{*{20}c} {\varTheta + \varTheta_{0} } & {\sqrt {\rho_{1} } F[I,I]} & {\sqrt {\rho_{1} - \rho_{2} } H[I,I]} \\ \bullet & { - L_{1} } & 0 \\ \bullet & \bullet & { - L_{3} } \\ \end{array} } \right] < 0 $$

(17)

$$ \left[ {\begin{array}{*{20}c} {\varTheta + \varTheta_{0} } & {\sqrt {\rho_{2} } F[I,\,I]} & {\sqrt {\rho_{1} - \rho_{2} } G[I,\,I]} \\ \bullet & { - L_{1} } & 0 \\ \bullet & \bullet & { - L_{2} } \\ \end{array} } \right] < 0 $$

(18)

$$ E_{u} L_{1} E_{u} + E_{l} L_{1} E_{l} - Z \le 0 $$

(19)

$$ E_{u} L_{2} E_{u} + E_{l} L_{2} E_{l} - Z_{1} \le 0,\;\;\;E_{u} L_{3} E_{u} + E_{l} L_{3} E_{l} - Z - Z_{1} \le 0 $$

(20)

$$ \left[ {\begin{array}{*{20}c} {\varXi_{11} } & {\varXi_{12} } & {\sqrt {\rho_{1} } \tilde{F}[I,I]} & {\sqrt {\rho_{2} - \rho_{1} } \tilde{H}[I,I]} & {\varXi_{15} } & {\varXi_{16} } & {\varXi_{17} } \\ \bullet & {\varXi_{22} } & 0 & 0 & {\varXi_{25} } & 0 & 0 \\ \bullet & \bullet & { - \tilde{L}_{1} } & 0 & 0 & 0 & 0 \\ \bullet & \bullet & \bullet & { - \tilde{L}_{3} } & 0 & 0 & 0 \\ \bullet & \bullet & \bullet & \bullet & { - \kappa_{1} I} & 0 & 0 \\ \bullet & \bullet & \bullet & \bullet & \bullet & { - \kappa_{1}^{ - 1} I} & 0 \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & { - I} \\ \end{array} } \right] < 0 $$

(21)

$$ \left[ {\begin{array}{*{20}c} {\varXi_{11} } & {\varXi_{12} } & {\sqrt {\rho_{2} } \tilde{F}[I,\,I]} & {\sqrt {\rho_{2} - \rho_{1} } \tilde{G}[I,\,I]} & {\varXi_{15} } & {\varXi_{16} } & {\varXi_{17} } \\ \bullet & {\varXi_{22} } & 0 & 0 & {\varXi_{25} } & 0 & 0 \\ \bullet & \bullet & { - \tilde{L}_{1} } & 0 & 0 & 0 & 0 \\ \bullet & \bullet & \bullet & { - \tilde{L}_{2} } & 0 & 0 & 0 \\ \bullet & \bullet & \bullet & \bullet & { - \kappa_{2} I} & 0 & 0 \\ \bullet & \bullet & \bullet & \bullet & \bullet & { - \kappa_{2}^{ - 1} I} & 0 \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & { - I} \\ \end{array} } \right] < 0 $$

(22)

where

$$ \begin{aligned} \varTheta = & \, [I_{2n} 0_{2n \times (6n + q)} ]^{T} P\tilde{M} + \tilde{M}^{T} P[I,\;0] + \tilde{M}_{z}^{T} \tilde{M}_{z} + F[I, - I,0] + [I, - I,0]^{T} F^{T} \\ & + {\text{diag}}\left\{ {Q + Q_{1} + Q_{2} ,(h - 1)Q, - Q_{1} , - Q_{2} , - \eta^{2} I_{q} } \right\} + G[0, - I,I,0] + [0, - I,I,0]^{T} G^{T} \\ & + H[0,I,0, - I,0] + [0,I,0, - I,0]^{T} H^{T} \\ \end{aligned} $$

$$ \begin{aligned} \varTheta_{0} & = \, [\tilde{M}^{T} ,\varphi_{0} \tilde{n}^{T} ](\rho_{2} Z + (\rho_{2} - \rho_{1} )Z_{1} )[\tilde{M}^{T} ,\varphi_{0} \tilde{n}^{T} ]^{T} , \\ \tilde{M} &= \, [M,M_{\tau } ,0_{2n \times 4n} ,B_{\delta w} ],\;\;\tilde{M}_{z} = [M_{z} ,0,B_{zw} ],\quad \tilde{N} = [N,N_{\tau } ,0_{2n \times (4n + q)} ], \\ E_{u}& = \, {\text{diag}}\{ I,\,0\} ,\quad E_{l} = {\text{diag}}\{ 0,\,I\} , \\ \end{aligned} $$

Given K _a , K _b , K _c and η > 0, the conditions of Theorem 1 are in terms of strict LMIs which could be easily solved using existing LMI solvers. Note that our purpose is to design LMI schemes to seek these feedback gains K _a , K _b and K _c . The maximum tolerant delay bound for $\rho_{2}$ can be searched and the minimum level of η can be computed simultaneously.

Theorem 2

Given the delay-interval bounds $\rho_{1} > 0,\;\rho_{2} > 0$ and η > 0 the closed-loop system (8) is EMS-η if there exist $n \times n$ matrices X > 0 and $Y\; > \;0,\;2n\; \times \;2n$ matrices $\tilde{Q} > 0,\;\tilde{Q}_{1} > 0$ and $\tilde{Q}_{2} > 0,\;4n\; \times \;4n$ matrices $\tilde{Z} > 0,\;\tilde{Z}_{1} > 0,\;\tilde{L}_{1} > 0,\;\tilde{L}_{1} > 0$ and $\tilde{L}_{1} > 0,\;(8n + q) \times 2n$ matrices $\tilde{F},\tilde{G}$ and $\tilde{H},\;n \times n$ matrix $\varUpsilon_{a} ,\;m \times n$ matrix $\varUpsilon_{b}$ and $n \times r$ matrix $\varUpsilon_{c}$, such that the following LMIs hold for some scalars $\kappa_{1} > 0$ and $\kappa_{2} > 0$ (21);

$$ E_{u} \tilde{L}_{1} E_{u} + E_{l} \tilde{L}_{1} E_{l} - \tilde{Z} < 0, $$

(23)

$$ E_{u} \tilde{L}_{2} E_{u} + E_{l} \tilde{L}_{2} E_{l} - \tilde{Z}_{1} < 0,\;\;E_{u} \tilde{L}_{3} E_{u} + E_{l} \tilde{L}_{3} E_{l} - \tilde{Z} - \tilde{Z}_{1} < 0 $$

(24)

$$ \begin{aligned} \varXi_{11} &= \left[ {\begin{array}{*{20}c} {YA + A^{T} Y} & {\varXi_{11a} } & 0 & 0 & 0 & {YB_{xw} } \\ {} & {\varXi_{11c} } & {(1 - \rho )\varUpsilon_{c} D} & {(1 - \rho )\varUpsilon_{c} D} & 0 & {XB_{xw} + \varUpsilon_{c} B_{yw} } \\ {} & * & 0 & 0 & 0 & 0 \\ {} & * & * & 0 & 0 & 0 \\ {} & * & * & * & {0_{4n} } & 0 \\ {} & * & * & * & * & {0_{q} } \\ \end{array} } \right] \\ & \quad + {\text{diag}}\left\{ {\tilde{Q} + \tilde{Q}_{1} + \tilde{Q}_{2} ,(h - 1)\tilde{Q}, - \tilde{Q}_{1} , - \tilde{Q}_{2} , - \eta^{2} I_{q} } \right\} \\ & \quad + \tilde{F}[I_{2n} , - I_{2n} ,0_{2n \times (4n + q)} ] + [I_{2n} , - I_{2n} ,0_{2n \times (4n + q)} ]^{T} \tilde{F}^{T} \\ & \quad + \tilde{G}[0_{2n} , - I_{2n} ,I_{2n} ,0_{2n \times (2n + q)} ] + [0_{2n} , - I_{2n} ,I_{2n} ,0_{2n \times (2n + q)} ]^{T} \tilde{G}^{T} \\ & \quad + \tilde{H}[0_{2n} ,I_{2n} ,0_{2n} , - I_{2n} ,0_{2n \times q} ] + [0_{2n} ,I_{2n} ,0_{2n} , - I_{2n} ,0_{2n \times q} ]^{T} \tilde{H}^{T} , \\ \varXi_{11a} &= A^{T} X + YA + \rho C^{T} \varUpsilon_{c}^{T} + \varUpsilon_{a}^{T} ,\;\varXi_{11c} = XA + A^{T} X + \rho \varUpsilon_{c} C + \rho C^{T} \varUpsilon_{c}^{T} \\ \varXi_{12}& = \left[ {\begin{array}{*{20}c} {A^{T} Y} & {\left( {A^{T} X + \rho C^{T} \varUpsilon_{c}^{T} + \varUpsilon_{a}^{T} } \right)} & 0 & {\varphi_{0} C^{T} \varUpsilon_{c}^{T} } \\ {A^{T} Y} & {\left( {A^{T} X + \rho C^{T} \varUpsilon_{c}^{T} } \right)} & 0 & {\varphi_{0} C^{T} \varUpsilon_{c}^{T} } \\ 0 & {(1 - \rho )D^{T} \varUpsilon_{c}^{T} } & 0 & { - \varphi_{0} D^{T} \varUpsilon_{c}^{T} } \\ 0 & {(1 - \rho )D^{T} \varUpsilon_{c}^{T} } & 0 & { - \varphi_{0} D^{T} \varUpsilon_{c}^{T} } \\ {0_{4n \times n} } & {0_{4n \times n} } & {0_{4n \times n} } & {0_{4n \times n} } \\ {B_{xw}^{T} Y} & {B_{xw}^{T} X + B_{yw}^{T} \varUpsilon_{c}^{T} } & 0 & 0 \\ \end{array} } \right] \\ \varXi_{22}& = - 2{\text{diag}}\left\{ {\left[ {\begin{array}{*{20}c} Y & Y \\ Y & X \\ \end{array} } \right],\;\left[ {\begin{array}{*{20}c} Y & Y \\ Y & X \\ \end{array} } \right]} \right\} + \rho_{2} \tilde{Z} + (\rho_{2} - \rho_{1} )\tilde{Z}_{1} , \\ \end{aligned} $$

$$ \varXi _{{15}} = \left[ {\begin{array}{*{20}c} Y \\ 0 \\ \end{array} } \right],\,\varXi _{{25}} = \left[ {\begin{array}{*{20}c} Y \\ 0 \\ \end{array} } \right],\,\varXi _{{16}} = \left[ {\begin{array}{*{20}c} {\varUpsilon _{b} ^{T} B^{T} } \\ 0 \\ \end{array} } \right],\,\varXi _{{17}} = \left[ {\begin{array}{*{20}c} {C_{z}^{T} + \varUpsilon _{b} ^{T} B_{z}^{T} } \\ {C_{z}^{T} } \\ 0 \\ {B_{{zw}}^{T} } \\ \end{array} } \right]$$

(25)

where $E_{u}$ and $E_{l}$ are as in Theorem 1. In this case, the feedback gains $K_{a} ,K_{b}$ and $K_{c}$ are given by

$$ K_{a} = U^{ - 1} (\varUpsilon_{a} - XB\varUpsilon_{b} )Y^{ - 1} W^{ - T} ,\;K_{b} = \varUpsilon_{b} Y^{ - 1} W^{ - T} ,\;K_{c} = U^{ - 1} \varUpsilon_{c} $$

(26)

where U and W are two invertible matrices satisfying $UW^{T} = I - XY^{ - 1} .$

Remark 1 Note that Theorem 2 provides an LMI method towards solving the matrix inequalities in (17–20), and hence presents controller designs of the form (7) to make the closed-loop system (8) EMS-η. The novelty of the result mainly lies in that an LMI design scheme is proposed for NCSs in continuous-time system settings with random measurements and time delays. Furthermore, the derivation is proceeded using appropriate Lyapunov functionals and matrix decoupling techniques.

4 Illustrative Example

To illustrate the theoretical developments, we consider a chemical reactor. The linearized model can be described by the following matrices:

$$ \begin{aligned} A = & \left[ {\begin{array}{*{20}c} { - 4.931} & { - 4.886} & {4.902} & 0 \\ { - 5.301} & { - 5.174} & { - 12.8} & {5.464} \\ {6.4} & {0.347} & { - 11.773} & { - 1.04} \\ 0 & {0.833} & {11.0} & { - 3.932} \\ \end{array} } \right],\;\;B^{t} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} } \right],\;B_{z} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ \end{array} } \right], \\ C = & D = \left[ {\begin{array}{*{20}c} {10} & 0 & 0 & 0 \\ \end{array} } \right],\;C_{z} = \left[ {\begin{array}{*{20}c} {0.8} & 1 & {0.1} & {0.2} \\ \end{array} } \right]^{T} ,\;B_{zw} = 0.4 \\ A_{z} = & \left[ {\begin{array}{*{20}c} {1.921} & {1.915} & 0 & {1.908} \\ \end{array} } \right],\;\;B_{xw} = \left[ {\begin{array}{*{20}c} {0.8} & 1 & {0.1} & {0.2} \\ \end{array} } \right]^{T} ,\;\;B_{yw} = 0.01 \\ \end{aligned} $$

Using the LMI toolbox in MATLAB, the ensuing results are summarized by:

$$ \begin{aligned} X = & \left[ {\begin{array}{*{20}c} {0.1448} & { - 0.0020} & {0.0005} & {0.0002} \\ { - 0.0020} & {0.1442} & { - 0.0005} & {0.0009} \\ {0.0005} & { - 0.0005} & {0.1420} & {0.0001} \\ {0.0002} & {0.0009} & {0.0001} & {0.1463} \\ \end{array} } \right], \\ Y = & \left[ {\begin{array}{*{20}c} {0.2142} & { - 0.0560} & {0.0421} & { - 0.0153} \\ { - 0.0560} & {0.0383} & { - 0.0138} & {0.0515} \\ {0.0421} & { - 0.0138} & {0.0292} & {0.0435} \\ { - 0.0153} & {0.0515} & {0.0435} & {0.2597} \\ \end{array} } \right], \\ \end{aligned} $$

With W = I and $U = I - XY^{ - 1} ,\;$ the corresponding feedback gains

$$ \begin{aligned} K_{a} = & \left[ {\begin{array}{*{20}c} {0.7573} & {0.7142} & {0.3973} & {0.8391} \\ {0.2138} & {8.2185} & {13.8882} & { - 3.4177} \\ \end{array} } \right], \\ K_{b} = & \left[ {\begin{array}{*{20}c} {0.3144} & { - 0.7983} & { - 3.8703} & {1.7806} \\ { - 0.6559} & {7.2776} & {14.8651} & { - 6.8895} \\ \end{array} } \right], \\ K_{c} = & \left[ {\begin{array}{*{20}c} {0.2634} & { - 0.1587} & { - 3.1912} & {1.5713} \\ { - 1.0803} & {8.5794} & {12.2875} & { - 3.9658} \\ \end{array} } \right] \\ \end{aligned} $$

Simulation of the closed-loop system is performed and the ensuing state trajectories are presented in Fig. 1. It is evident that the the closed-loop system is EMSS-η.

Fig. 1

Closed-loop state trajectories

5 Conclusion

An LMI method has been presented for observer-based $H_{\infty }$ control of NCSs in continuous-time system settings with random measurements and probabilistic time delays. Improved schemes have been shown for the design method. It has been established that these conditions reduce the conservatism by considering not only the range of the time delays, but also the probability distribution of their variation. A numerical simulation example has been presented to show the merits and advantages of the proposed techniques.

Appendix

Proof of Theorem 1

The proof is twofold: we first choose a functional J of the form (13) to show that the $H_{\infty }$ performance requirement (12) is satisfied, and then use the Lyapunov functional V of the form (15) to prove the EMS property. Denote

$$ \chi (t): = [\delta (t)^{T} ,\delta_{\tau }^{T} ,\delta_{1}^{T} ,\delta_{2}^{T} ,w(t)^{T} ]^{T} ,\;\;\delta_{\tau } : = \delta (t - \delta_{t} ),\;\;\delta_{i} : = \delta (t - \delta_{i} ),\;\;i = 1,\,2. $$

(27)

From the Newton–Leibniz formula $0 = \delta (t) - \delta_{\tau } - \int\nolimits_{t - \tau (t)}^{t} {\dot{\delta }(s)ds}$, we have that

$$ \begin{aligned} \psi_{1} (t): &=\, 2\chi (t)^{T} F\left[ {\delta (t) - \delta_{\tau } - \int\limits_{t - \tau (t)}^{t} {\dot{\delta }(s)ds} } \right] = 0, \\ \psi_{2} (t):&= \,2\chi (t)^{T} G\left[ {\delta_{1} - \delta_{\tau } - \int\limits_{t - \tau (t)}^{{t - \rho_{1} }} {\dot{\delta }(s)ds} } \right] = 0, \\ \psi_{3} (t):&= \,2\chi (t)^{T} G\left[ {\delta_{\tau } - \delta_{2} - \int\limits_{{t - \rho_{2} }}^{t - \tau (t)}{\dot{\delta }(s)ds} } \right] = 0 \\ \end{aligned}$$

(28)

hold for any $(8n + q) \times 2n$ matrices F, G and H. Let the functional $J(\delta_{t} ;t)$ be chosen as in (13), then, from (16), $\fancyscript{L} J$ for the evolution of J is given by [20]

$$\begin{aligned} \fancyscript{L}J(\delta_{t} ;t) & = 2\delta (t)^{T}Pf(\delta ,t) + \delta (t)^{T} (Q + Q_{1} + Q_{2} )\delta (t) \\&\quad - (1 - \dot{\tau }_{t} )\delta_{\tau }^{T} Q\delta_{\tau } -\sum\limits_{i = 1}^{2} \delta_{i}^{T} Q_{i} \delta_{i} \\ &\quad+ \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0}g(\delta ,s)} \\ \end{array} } \right]^{T} (\rho_{2} Z + (\rho_{2} -\rho_{1} )Z_{1} )\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\{\varphi_{0} g(\delta ,s)} \\ \end{array} } \right] -\int\limits_{{t -\rho_{2} }}^{t} \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\{\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} Z\left[{\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)}\\ \end{array} } \right]ds \\ & \quad- \int\limits_{{t + \rho_{2} }}^{{t +\rho_{1} }} \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\{\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} Z_{1} \left[{\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)}\\ \end{array} } \right]ds \\ & = 2\delta (t)^{T} Pf(\delta ,t) +\delta (t)^{T} (Q + Q_{1} + Q_{2} )\delta (t) - (1 - \dot{\tau }_{t})\delta_{\tau }^{T} Q\delta_{\tau } - \sum\limits_{i = 1}^{2}\delta_{i}^{T} Q_{i} \delta_{i} \\ &\quad + \psi_{1} (t) + \psi_{2}(t) +\psi_{3} (t) + \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\{\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} (\rho_{2} Z+ (\rho_{2} - \rho_{1} )Z_{1} )\left[ {\begin{array}{*{20}c}{f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} }\right] \\ &\quad - \int\limits_{t - \tau (t)}^{t} \left[{\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)}\\ \end{array} }\right]^{T} Z\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\{\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]ds -\int\limits_{t - \tau (t)}^{{t - \rho_{1} }} \left[{\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)}\\ \end{array} } \right]^{T} Z_{1} \left[ {\begin{array}{*{20}c}{f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} }\right]ds \\ &\quad - \int_{{t - \rho_{2}}}^{t - \tau (t)} \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\{\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} (Z + Z_{1})\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0}g(\delta ,s)} \\ \end{array} } \right]ds \\ \end{aligned}$$

(29)

Note that, in $\psi_{1} (t)$, the following inequality holds for any $4n \times 4n$ matrix $L > 0,\,thus:$

$$ \begin{aligned} - 2\chi (t)^{T} F\int\limits_{t - \tau (t)}^{t} {\dot{\delta }(s)ds} &= - 2\chi (t)^{T} F\left[ {I_{2n} ,I_{2n} } \right]\int\limits_{t - \tau (t)}^{t} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {(\varphi (s) - \rho )g(\delta ,s)} \\ \end{array} } \right]ds} \\ & \le \tau (t)\chi (t)^{T} F\left[ {I_{2n} ,I_{2n} } \right]L_{1}^{ - 1} \left[ {I_{2n} ,I_{2n} } \right]^{T} F^{T} \chi (t) \\ & \quad + \int\limits_{t - \tau (t)}^{t} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {(\varphi (s) - \rho )g(\delta ,s)} \\ \end{array} } \right]^{T} L_{1} \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {(\varphi (s) - \rho )g(\delta ,s)} \\ \end{array} } \right]ds} \\ \end{aligned} $$

Similarly, in $\psi_{i} (t)\;(i = 2,3)$, the following inequalities hold for any $4n\; \times \;4n$ matrices $L_{i} > 0 \text{:}$

$$ \begin{aligned} & - 2\chi (t)^{T} G\int\limits_{t - \tau (t)}^{{t - \rho_{1} }} {\dot{\delta }(s)ds} \,\leq \, (\tau (t) - \rho_{1} )\chi (t)^{T} G\left[ {I,\;I} \right]L_{2}^{ - 1} \left[ {I,\;I} \right]^{T} G^{T} \chi (t) \\ &\quad + \int\limits_{t - \tau (t)}^{{t - \rho_{1} }} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {(\varphi (s) - \rho )g(\delta ,s)} \\ \end{array} } \right]^{T} L_{2} \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {(\varphi (s) - \rho )g(\delta ,s)} \\ \end{array} } \right]ds} \\ \end{aligned} $$

and

$$ \begin{aligned} & - 2\chi (t)^{T} H\int\limits_{t - \tau (t)}^{{t - \rho_{1} }} {\dot{\delta }(s)ds} \,\leq\, (\rho_{2} - \tau (t))\chi (t)^{T} H\left[ {I,\;I} \right]L_{3}^{ - 1} \left[ {I,\;I} \right]^{T} H^{T} \chi (t) \\ & \quad+ \int\limits_{{t - \rho_{2} }}^{t - \tau (t)} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {(\varphi (s) - \rho )g(\delta ,s)} \\ \end{array} } \right]^{T} L_{3} \left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {(\varphi (s) - \rho )g(\delta ,s)} \\ \end{array} } \right]ds} \\ \end{aligned} $$

Considering (6, 8, and 10) and taking the expectation on (29), we have

$$ \begin{aligned} I\;E&\{ {\fancyscript{L}}V(\delta_{t} ,t) + {\parallel }z(t){\parallel }^{2} - \eta^{2} {\parallel }w(t){\parallel }^{2} \} \leq I\;E\{ \chi (t)^{T} (\varTheta + \varTheta_{0} \\ & + \tau (t)\varTheta_{1} + (\tau (t) - \rho_{1} )\varTheta_{2} + (\rho_{2} - \tau (t))\varTheta_{3} )\chi (t)\} + \psi_{4} (t) \\ \end{aligned} $$

(30)

and

$$ \begin{aligned} \varTheta_{1} &= F\left[ {I,\;I} \right]L_{1}^{ - 1} \left[ {I,\;I} \right]^{T} F^{T} ,\;\;\varTheta_{2} = G\left[ {I,\;I} \right]L_{2}^{ - 1} \left[ {I,\;I} \right]^{T} G^{T} , \\ \varTheta_{3} &= H\left[ {I,\;I} \right]L_{3}^{ - 1} \left[ {I,\;I} \right]^{T} H^{T} , \\ \psi_{4} (t) = & \int\limits_{t - \tau (t)}^{t} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} \left( {E_{u} L_{1} E_{u} + E_{l} L_{1} E_{l} - Z} \right)\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]ds} \\ & + \int\limits_{t - \tau (t)}^{{t - \rho_{1} }} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} \left( {E_{u} L_{2} E_{u} + E_{l} L_{2} E_{l} - Z_{1} } \right)\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]ds} \\ & + \int\limits_{{t - \rho_{2} }}^{t - \tau (t)} {\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]^{T} \left( {E_{u} L_{3} E_{u} + E_{l} L_{3} E_{l} - Z - Z_{1} } \right)\left[ {\begin{array}{*{20}c} {f(\delta ,s)} \\ {\varphi_{0} g(\delta ,s)} \\ \end{array} } \right]ds} \\ \end{aligned} $$

Applying the Schur complement, conditions (17) and (18) are equivalent to

$$ \tilde{\varTheta }_{1} = \varTheta + \varTheta_{0} + \rho_{1} \varTheta_{1} + (\rho_{2} - \rho_{1} )\varTheta_{3} < 0, $$

(31)

$$ \tilde{\varTheta }_{2} = \varTheta + \varTheta_{0} + \rho_{2} \varTheta_{1} + (\rho_{2} - \rho_{1} )\varTheta_{2} < 0, $$

(32)

From (19–20, 31, 32) and Lemma 2, we deduce from (30) that

$$ {\mathbf{E}}{\kern 1pt} \{ {\fancyscript{L}}J(\delta_{t} ,t) + {\parallel }z(t){\parallel }^{2} - \eta^{2} {\parallel }w(t){\parallel }^{2} \} \le \hbox{max} \{ \lambda_{max} (\tilde{\varTheta }_{1} ),\lambda_{max} (\tilde{\varTheta }_{2} )\} \;\;{\mathbf{E}}{\kern 1pt} \{ {\parallel }\chi {\parallel }^{2} \} \le 0$$

(33)

Under zero-initial conditions and noticing $J(\delta_{T} ;T) \ge 0$ for any T > 0, integrating (33) from 0 to ∞ yields that the $H_{\infty }$ performance requirement (12) is satisfied. With a procedure similar to the above, we can arrive under the given conditions and by virtue of Lemma 2 that,

$$ {\mathbf{E}}{\kern 1pt} \{ {\fancyscript{L}}V(\delta_{t} ,t)\} \le \hbox{max} \{\lambda_{max} (\tilde{\varTheta }_{1} ),\lambda_{max} (\tilde{\varTheta }_{2} )\} \;\;{\mathbf{E}}\{{\parallel}\delta (t){\parallel}^{2} \} $$

Hence, system (8) is EMS from Lemma 1.

Proof of Theorem 2

It can be seen from (21) or (22) that

$$ \left[ {\begin{array}{*{20}c} Y & Y \\ Y & X \\ \end{array} } \right] > 0 $$

which gives $XY > 0,$ implying that $I - - XY^{ - 1}$ is invertible. Now let U and W be any invertible matrices satisfying $UW^{T} = I - XY^{ - 1}$. Choose

$$ P = \left[ {\begin{array}{*{20}l} X & U \\ {U^{T} } & * \\ \end{array} } \right] > 0,\;P^{ - 1} = \left[ {\begin{array}{*{20}l} {Y^{ - 1} } & W \\ {W^{T} } & * \\ \end{array} } \right] > 0 $$

(34)

where each ellipsis ${*}$ denotes a positive definite matrix block that will not influence the subsequent development (of course it makes $PP^{ - 1} = I$). In the sequel, we show that if (21, 24) are satisfied, then (17, 20) hold with $P > 0$ chosen as in (39), and thus the result follows immediately from Theorem 1. Define

$$ S = \left[ {\begin{array}{*{20}c} I & I \\ {W^{T} Y} & 0 \\ \end{array} } \right] $$

(35)

which is invertible and produces

$$ S^{T} P = \left[ {\begin{array}{*{20}c} Y & 0 \\ X & U \\ \end{array} } \right],\quad S^{T} PS = \left[ {\begin{array}{*{20}c} Y & Y \\ Y & X \\ \end{array} } \right]. $$

(36)

We first show that (21) implies (17). By Schur complement, the matrix inequality (17) holds if and only if (37)

$$ \begin{aligned} & \left[ {\begin{array}{*{20}c} \varTheta & {[\tilde{M}^{T} ,\varphi_{0} \tilde{N}^{T} ]{\text{diag}}\left\{ {P,P} \right\}} & {\sqrt {\rho_{1} } F[I,I]} & {\sqrt {\rho_{2} - \rho_{1} } H[I,I]} \\ \bullet & { - \varPi } & 0 & 0 \\ \bullet & \bullet & { - L_{1} } & 0 \\ \bullet & \bullet & \bullet & { - L_{3} } \\ \end{array} } \right] < 0 \\ & \varPi = {\text{diag}}\left\{ {P,P} \right\}(\rho_{2} Z + (\rho_{2} - \rho_{1} )Z_{1} )^{ - 1} {\text{diag}}\left\{ {P,P} \right\} \\ \end{aligned} $$

(37)

In view of

$$\begin{aligned} & \left( {\rho_{2} Z + (\rho_{2} - \rho_{1} )Z_{1} - {\text{diag}}\{ P,P\} } \right)(\rho_{2} Z + (\rho_{2} - \rho_{1} )Z_{1} )^{ - 1} \\ & times \left( {\rho_{2} Z + (\rho_{2} - \rho_{1} )Z_{1} - {\text{diag}}\{ P,P\} } \right) > 0 \\ \end{aligned}$$

(38)

we obtain

$$\begin{aligned} & - {\text{diag}}\left\{ {P,P} \right\}(\rho _{2} Z + (\rho _{2} - \rho _{1} )Z_{1} )^{{ - 1}} {\text{diag}}\left\{ {P,P} \right\} \le - 2{\text{diag}}\left\{ {P,P} \right\} \\ &+ \rho _{2} Z + (\rho _{2} - \rho _{1} )Z_{1} \\ \end{aligned}$$

(39)

we have that (37) holds if (40) holds

$$ \left[ {\begin{array}{*{20}c} \varTheta & {[\tilde{M}^{T} ,\varphi_{0} \tilde{N}^{T} ]{\text{diag}}\left\{ {P,P} \right\}} & {\sqrt {\rho_{1} } F[I,I]} & {\sqrt {\rho_{2} - \rho_{1} } H[I,I]} \\ \bullet & { - 2{\text{diag}}\left\{ {P,P} \right\} + \rho_{2} Z + (\rho_{2} - \rho_{1} )Z_{1} } & 0 & 0 \\ \bullet & \bullet & { - L_{1} } & 0 \\ \bullet & \bullet & \bullet & { - L_{3} } \\ \end{array} } \right] < 0 $$

(40)

Now, applying the congruence transformation $\text{diag}\left\{ {S,S,S,S,I_{q} ,S,S,S,S,S,S} \right\}$ to (40) and setting

$$ \begin{aligned} \tilde{Q} &= S^{T} QS,\quad \tilde{Q}_{1} = S^{T} Q_{1} S,\quad \tilde{Q}_{2} = S^{T} Q_{2} S, \\ \tilde{Z}& = {\text{diag}}\left\{ {S,S} \right\}^{T} Z{\text{diag}}\left\{ {S,S} \right\},\;\;\tilde{Z}_{1} = {\text{diag}}\{ S,S\}^{T} Z_{1} {\text{diag}}\left\{ {S,S} \right\}, \\ \tilde{L}_{i} &= {\text{diag}}\left\{ {S,S} \right\}^{T} L_{i} {\text{diag}}\{ S,S\} ,\quad i = 1,2,3 \\ \tilde{F} &= {\text{diag}}\left\{ {S,S,S,S} \right\}^{T} FS,\;\;\tilde{G} = {\text{diag}}\left\{ {S,S,S,S} \right\}^{T} GS, \\ \tilde{H} &= {\text{diag}}\left\{ {S,S,S,S} \right\}^{T} HS,\;\;\varUpsilon_{a} = XBK_{b} W^{T} Y + UK_{a} W^{T} Y, \\ \varUpsilon_{b} &= K_{b} W^{T} Y,\;\;\varUpsilon_{c} = UK_{c} \\ \end{aligned} $$

(41)

we obtain that (40) is equivalent to

$$ \tilde{\varXi } + \tilde{Y}\tilde{K} + \tilde{K}^{T} \tilde{Y}^{T} \; < \;0, $$

(42)

$$ \begin{aligned} \tilde{\varXi } = & \left[ {\begin{array}{*{20}c} {\varXi_{11} + \varXi_{17}^{T} \varXi_{17} } & {\varXi_{12} } & {\sqrt {\rho_{1} } \tilde{F}[I,I]} & {\sqrt {\rho_{2} - \rho_{1} } \tilde{H}[I,I]} \\ \bullet & {\varXi_{22} } & 0 & 0 \\ \bullet & \bullet & { - \tilde{L}_{1} } & 0 \\ \bullet & \bullet & \bullet & { - \tilde{L}_{3} } \\ \end{array} } \right] \\ \tilde{Y} = & \left[ {\varXi_{15}^{T} ,\varXi_{25}^{T} ,0_{n \times 8n} } \right]^{T} ,\quad \tilde{K} = \left[ {\varXi_{16}^{T} ,0_{n \times 12n} } \right] \\ \end{aligned} $$

Inequality (42) holds if the following is true for any $\kappa_{1} > 0,$

$$ \tilde{\varXi } + \kappa_{1}^{ - 1} \tilde{Y}\tilde{Y}^{T} + \kappa_{1} \tilde{K}^{T} \tilde{K} < 0, $$

(43)

which is equivalent to

$$ \left[ {\begin{array}{*{20}c} {\tilde{\varXi }} & {\tilde{Y}} & {\tilde{K}^{T} } \\ {\tilde{Y}^{T} } & { - \kappa_{1} I_{n} } & 0 \\ {\tilde{K}} & 0 & { - \kappa_{1}^{ - 1} I_{n} } \\ \end{array} } \right] < 0 $$

(44)

The above inequality is, by Schur complement again, exactly that of (21), and we conclude that this implies (17).

Next we show that (22) implies (18). This can be done by using a procedure analogous to the above. As for the verification of other inequalities, applying the congruence transformation diag{S, S} to (19, 20) and setting matrix variables as in (41), it is seen that (19, 20) are equivalent to (23, 24). So far, we have proven that (21– 24) ensure (17–20) and thus the closed-loop system (8) is EMS-η. In this case, from (41), the feedback gains are computed as in (26). This completes the proof.