Mean-Square Input-Output Stability and Stabilizability of a Networked Control System with Random Channel Induced Delays

Networked feedback control systems are known as spatially distributed control systems in which signals are exchanged over communication network [1]. In the past few decades, a huge amount of research interests has been attracted on the analysis and design of these systems due to great advantages in networked control systems, such as ease for installation and maintenance, reduced system wiring, and resources sharing etc. Typical application examples of networked control systems can be found in a broad range of areas such as: automotive industry [2], autonomous underwater vehicles and unmanned air vehicles [3, 4], and even remote surgery [5] and big data [6]. Despite of the advantages, networked control systems also pose challenging problems arising from unreliability in data transmission, caused by channel induced delays (i.e., data transmission times), packet dropout, coding error etc. Due to the fact that data are transmitted in network by data packet, channel induced delays and data packet dropout that occur in data exchanging between components of feedback systems over network are the most common phenomena in networked systems. The channel induced delays and packet dropout could seriously degrade performance of feedback control systems, and even destabilize feedback systems when these issues are not carefully considered in system design (see, e.g., [7, 8] and [9]). To cope with channel induced delays, great efforts have been made in modeling, stability analysis, and controller design of networked feedback systems (for example see [7, 8] and the references therein).

Channel induced delays in networked systems could be constant, time-varying, and random delays, which are basically dependent on network protocol, channel quality etc. In [10, 11, 12], the authors studied the asymptotic stability of networked linear feedback systems in which message is transmitted over network with a constant transmission time (i.e. a constant channel induced delay). The networked systems were modeled as discrete-time linear systems augmented from discretized plants, delay channel models and control laws. In particular, the channel induced delay was modeled as a parameter of the augmented linear systems’ matrices. Stability criteria were obtained for the networked systems in terms of the discrete-time linear system models. These augmented models were also used to study the asymptotic stability and stabilization design for networked feedback systems with time-varying delays. Several stability criteria and stabilization design results were obtained by using switch system approaches (see for example [13, 14, 15, 16]). Alternatively, time-varying channel induced delays were modeled as continuous-time plants’ input delays (see e.g., [17, 18, 19]). Lyapunov-Krasovskii functionals were used in stability analysis and stabilization design for networked feedback systems. Various linear matrix inequality (LMI) based methods were developed. Moreover, to cope with data disorder occurred in transmission, time-stamp scheme and logic zero-order hold were developed for the networked feedback systems [20, 21]. In general, the aims of aforementioned papers are to find criteria that networked feedback systems are stable when channel induced delays belong to given regions, or to find upper bounds of channel induced delays under which the stability of the networked feedback systems are preserved. This leads to certain conservativeness in stability analyses and stabilization design for networked feedback systems with random channel induced delays.

Since channel induced delays and packet dropouts usually exhibit random characteristics, random channel induced delay and packet dropout models are widely used in stability analysis and control design for networked feedback systems (see for example [22, 20, 23, 24, 25]). In [24, 25], applying Markov chains, the authors modeled networked systems with random delays and packet dropouts as discrete-time Markov jump linear systems. Necessary and sufficient conditions were obtained for the mean-square stability of the networked feedback systems. Then, the mean-square stabilization was studied for these systems and LMI based control design approaches were developed. It turned out that this stabilization problem is a non-convex problem. Certain conservativeness may not be avoidable in this type of results. On the other hand, channel induced delays were modeled as random input delays of plant/controllers, and the sequences of the input delays were assumed to be independent and identically distributed (i.i.d.) random processes in [22, 20]. It was assumed in these works that the probability density functions of the random channel induced delays are known and all the channel induced delays in networked feedback systems are smaller than the systems’ sampling intervals. A suboptimal linear controller was designed for a networked feedback system with a fixed sampling interval in [20]. The reference [22] studied a networked system with a time-varying sampling interval. LMI based sufficient conditions were presented for the mean-square stability of the system. However, for the case when channel induced delays are longer than the systems’ sampling intervals, the channel induced delays may not exactly be input delays of plants since more than one data may be received simultaneously or none data could be received at one sampling instant. In [26, 23, 27, 28], channel induced delays (or data transmission times) which are longer than the systems’ sampling intervals were modeled as an i.i.d. random process with a known probability mass function (PMF). State estimation problem was studied for these networked systems and optimal state estimation algorithms were developed in [27, 28]. The stabilization problem was studied for a nonlinear networked feedback system and a sufficient condition was obtained for the stability of the networked feedback system in [23].

In this work, we focus on a single-input single-output linear time-invariant (LTI) networked feedback system in which the control signal is transmitted over a communication channel with random channel induced delays. The networked feedback system is a discrete-time system with a fixed sampling interval and channel induced delays are integral multiples of this sampling interval. Our goal is to fully understand how the interaction between stochastic features of the random delays and characteristics of the networked feedback system affects the system’s stability and limits the stabilizability of the system. With these purposes, we adopt the random process model studied in [26, 23, 27, 28] to describe the random channel induced delays. At the channel terminal, a linear receiving strategy which is a linear combination with given weights of data received at one sampling time is adopted. The new signal generated by the linear receiving strategy is applied to the plant of the networked feedback system as a control signal. Then, the channel uncertainty caused by the random channel induced delays is defined by the random impulse response of the communication channel and the first-order statistics of the impulse response. The input-output relation of the channel uncertainty is established in terms of the spectral density of the uncertainty’s impulse response. To give a precise description for a channel relative deviation induced by the channel uncertainty, a concept of so-called frequency response of variation is introduced in frequency domain. A necessary and sufficient condition is presented for the mean-square input-output stability of the networked feedback system. It is a new version of the small gain theorem for networked feedback systems with random channel induced delays. It is also found that for the mean-square input-output stability and stabilizability problems, the channel uncertainty caused by the random channel induced delays is equivalent to an additive noise in an additive noise channel with a signal-to-noise ratio constraint (see [29] and [30] for more details in the mean-square stability of networked feedback systems over additive noise channels with a signal-to-noise ratio constraint). After then, the mean-square input-output stabilizability via output feedback is studied for the networked feedback system. In particular, a necessary and sufficient condition is found for the mean-square input-output stabilizability of the system when its plant is minimum phase. It precisely describes the connection between the mean-square input-output stabilizability, the frequency response of variation and the unstable poles of the plant in the feedback system. Furthermore, it turns out that the interaction between the frequency response of variation and the unstable poles plays a critical role to the mean-square stabilizability of the networked feedback system.

The remainder of this paper is organized as follows. In Section 2, the random channel induced delays and related channel uncertainty are modeled by using the PMF of the delays. The impulse responses of the channel and channel uncertainty are given. The problems under study are formulated. Section 3 presents the input-output relation of the channel uncertainty based on the spectral density of its impulse response. A necessary and sufficient condition of the mean-square input-output stability is obtained for the networked feedback system. In Section 4, the mean-square input-output stabilizability is studied for the networked feedback system when its plant is minimum phase. An analytic expression is obtained for the necessary and sufficient condition of the mean-square input-output stabilizability. The connection between the result in this work and the existing results is discussed. Section 5 illustrates some numerical examples and Section 6 concludes the paper.

The notations used in this paper is mostly standard. The complex conjugate transpose of any matrix $A$ is denoted by $A^{*}$. When $A$ is square and invertible, its inverse and inverse conjugate transpose are denoted by $A^{-1}$ and $A^{-*}$, respectively. For any transfer function $G(z)$, we represent a state-space realization of $G(z)$ by $G(z)=\left[\begin{array}[]{c|c}A&B\\ \hline\cr C&D\end{array}\right]$. Let the open unit disc be denoted by $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, the closed unit disc by $\bar{\mathbb{D}}:=\{z\in\bar{\mathbb{C}}:|z|\leq 1\}$, the unit circle by $\partial\mathbb{D}$, and the complements of $\mathbb{D}$ and $\bar{\mathbb{D}}$ by $\mathbb{D}^{c}$ and $\bar{\mathbb{D}}^{c}$, respectively. In this work, the Hardy space $\mathcal{H}_{2}$ consists of scalar-valued analytic function $F$ in $\bar{\mathbb{D}}^{c}$ such that

The orthogonal complement of $\mathcal{H}_{2}$ is given by

Define also the space $\mathcal{RH}_{\infty}$ as the set of all proper stable rational transfer functions. Furthermore, $\operatorname*{\mathrm{\bf E}}\{\cdot\}$ denotes the expectation operator of a random variable. The set of real numbers is denoted by $\mathbb{R}$.

Consider a canonical structure of a discrete-time LTI networked feedback system as depicted in Fig. 1. Here, the plant $P$ is an LTI system and its transfer function $P(z)$ is assumed to be strictly proper. The controller $K$ is an LTI controller.

The signal $y(k)\in\mathbb{R}$ is the measurement of the plant $P$, $u(k)\in\mathbb{R}$ is the control signal generated by the linear controller $K$, and $v(k)$ is the external input. An unreliable communication channel is placed in the path from the controller to the receiver. Here, the unreliable features under study are random channel induced delays and packet dropout. Denote the channel induced delay for the control signal $u(k)$ (i.e., the transmission time spent on transmitting $u(k)$ over the communication channel) by $\tau_{k}$. Thus, the signal $u(k)$ sent at the time $k$ arrives at its destination at the time $k+\tau_{k}$. The channel induced delay $\tau_{k}$ is assumed to be a random variable with nonnegative integer values from a bounded set $\mathcal{D}=\{0,1,2,\cdots,\bar{\tau}-1,\bar{\tau}\}$. All transmitted data are with time stamps. The data whose channel induced delays are greater than $\bar{\tau}$ are discarded at the receiver. ${\bf\eta}(k)$ is a collection of data, which has $\bar{\tau}+1$ entries and includes all data received at time $k$. Now, we use Kronecker delta function $\delta(k)$ to characterize all entries belonging to ${\bf\eta}(k)$,

Let ${\bf\eta}(k)=\left\{\eta_{0}(k),\eta_{1}(k),\cdots,\eta_{\bar{\tau}}(k)\right\}$ and $\eta_{i}(k)=\delta(\tau_{k-i}-i)u(k-i),\leavevmode\nobreak\ i\in\mathcal{D}$. The function $\delta(\tau_{k-i}-i)=1$ indicates that the signal $u(k-i)$ arrives at its destination through the channel at time $k$ and $\eta_{i}(k)=u(k-i)$, otherwise it means that $u(k-i)$ is not received at time $k$ and $\eta_{i}(k)=0$. It is assumed that the receiver at the terminal of the channel has limited computation capability. This capability allows the receiver to generate the signal $u_{d}(k)\in\mathbb{R}$ as its output based on the received data and the time stamps associated with the data. A linear combination of the received data $\{\eta_{0}(k),\cdots,\eta_{\bar{\tau}}(k)\}$ can be taken as the output of the receiver:

where the weights $\alpha_{0},\alpha_{1},\ldots,\alpha_{\bar{\tau}}$ are assigned to the received data, respectively, according to the delay steps of the data. We refer to the receiver given by (1) as linear receiver or linear receiving strategy. Note that the linear receiving strategy is a general case of the zero-input strategy [31]. Without loss of generality, we assume that the initial time is at $k=0$ and the system is at rest at initial time.

According to the discussion above, the networked feedback system in Fig. 1 with the linear receiver is re-diagrammed as that shown in Fig. 2, wherein $\operatorname{\mathbf{q}}^{-1}$ is the unit delay operator.

One can see from this block-diagram that the block cascaded by the channel and the receiver, referred to as a transmission block, is a linear system with a random finite impulse response. Its input-output relation is given by (1). Since the initial time is assumed to be zero, without loss generality, we consider the random impulse response $h(k,n)$ of the transmission block to a unit impulse input applied to the channel at any time $n\geq 0$, which is given by

One can see from (2) that the random impulse response $h(k,n)$ is determined by the instant $n$ and the random sequence $\{\alpha_{0}\delta(\tau_{n}),\alpha_{1}\delta(\tau_{n}-1),\cdots,\alpha_{\bar{\tau}}\delta(\tau_{n}-\bar{\tau})\}$. For a given realization of $\tau_{n}$, at most one entry is not equal to zero in the random sequence. The input-output relation of the transmission block is rewritten as

We impose the following assumptions throughout the paper.

Since the random sequence $\{\alpha_{0}\delta(\tau_{n}),\alpha_{1}\delta(\tau_{n}-1),\cdots,\alpha_{\bar{\tau}}\delta(\tau_{n}-\bar{\tau})\}$ is dependent on the random variable $\tau_{n}$ which is an entry of an i.i.d. process, the mean of each entry in this sequence is obtained from the PMF of $\tau_{n}$, i.e.,

Define the mean channel as

Thus, it holds that

Subsequently, the transmission block is divided into two parts: One is the mean channel $H(z)$ and the other is a zero-mean channel uncertainty denoted by $\Omega$. Denote the response of the latter part by $\omega(k,n)$ to the unit impulse input applied to the channel at time $n\geq 0$. This impulse response is given as below:

Accordingly, the receiver output $u_{d}(k)$ is the summation of the outputs of $H$ and $\Omega$ when considering $u(k)$ as their inputs, i.e.,

where

As a result, the system in Fig. 2 can be re-diagramed as a stochastic system shown in Fig. 3. The structure of this system is similar to that studied in literatures for networked feedback systems over fading channels (see for example [32] and [33]). In the literatures, the channel uncertainties under study are white noise processes, thus the mean channels are constants and the channel uncertainties are zero-mean white noise processes. But, in this work, the mean channel $H(z)$ and channel uncertainty $\Omega$ are linear systems with the time-invariant finite impulse response $\left\{\alpha_{0}p_{0},\cdots,\alpha_{\bar{\tau}}p_{\bar{\tau}}\right\}$ and the random finite impulse response $\left\{\omega(n,n),\cdots,\omega(n+\bar{\tau},n)\right\}$, respectively.

In Fig. 3, denote the system from $d$ to $u$ without considering the channel uncertainty $\Omega$ by $G(z)$, referred to as the nominal system, which is given by

Then the whole system is an interconnection of the nominal system $G(z)$ and the channel uncertainty $\Omega$, as shown in Fig. 4.

It is well-known that there exists a controller $K$ to internally stabilize the nominal feedback system $G$ for any stabilizable and detectable LTI plant $P$ if and only if there is not unstable pole-zero cancelation between the mean channel $H$ and the plant $P$. That is, the following assumption is necessary.

To avoid any possible unstable pole-zero cancelation, the weights $\alpha_{0},\alpha_{1}$, $\cdots,\alpha_{\bar{\tau}}$ in the linear receiving strategy can also be selected so that the mean channel $H$ is minimum phase. A numerical example is presented in Section 5. Since the controller-receiver co-design is a very difficult task in general, a common setup in most of literatures is that the controller is designed based on a given receiving strategy, e.g., zero-input strategy or hold-input strategy for packet dropout problem [32, 31]. In this work, we restrict ourselves to the fixed weights $\{\alpha_{0},\cdots,\alpha_{\bar{\tau}}\}$ of the linear receiving strategy such that $H$ is minimum phase. Throughout this paper, we concentrate on the mean-square input-output stability and stabilizability defined next.

Let the set of all proper controllers internally stabilizing $G(z)$ be $\mathcal{K}$.

In this section, we study the criterion of mean-square input-output stability for the networked feedback system with a given output feedback controller $K\in{\cal K}$. The frequency variable $z$ will be omitted whenever no confusion is caused.

It is shown in preceding section that a networked feedback system with a random channel induced delay and a linear receiver is modeled as a stochastic system shown in Fig. 3. Intuitively, the mean-square input-output stability of the system is determined by the interaction between the nominal feedback system $G$ and the zero-mean channel uncertainty $\Omega$. To study the interaction between $G$ and $\Omega$, the stochastic properties of the channel uncertainty $\Omega$ whose impulse response is given in (6) are studied.

Now the second order stochastic properties of $\Omega$ are studied. For any given instant $n\geq-\bar{\tau}$, let the autocorrelation of the subsequence $\{\omega(k,n):-\infty<k<\infty\}$ be given by

Note the fact that $\omega(k,n)\equiv 0$ for $k<n$ and $k>n+\bar{\tau}$. For the case $l=0$, only the terms with $n\leq k\leq n+\bar{\tau}$ in the summation of (13) may not be equal to zero. By letting $k_{1}=k_{2}=k$, $i=k-n$ and applying Lemma 1.2 into (13), we obtain that

According to Lemma 1.3 and (13), letting $k_{1}=k$, $k_{2}=k+l$, $i_{1}=k-n$ and $i_{2}=k+l-n$ yields

It holds for $l>\bar{\tau}$ that $\omega(k,n)\omega(k+l,n)\equiv 0$. Hence, $r(l)\equiv 0$ for $l>\bar{\tau}$.

In the case when $l<0$, note the fact that $\omega(k+l,n)\equiv 0$ for any $k+l<n$. It is verified by (13) that $r(l)=r(-l).$

Subsequently, define the energy spectral density of the channel uncertainty $\Omega$ as follows:

Define the autocorrelation of the sequence $\{d(0),d(1),\cdots\}$ by

It follows from (21) that

The power spectral density of $d(k)$ is as follows:

Denote the $i$th-component of $d(k)$ in (9) by $d_{i}(k)$, $i\in{\mathcal{D}}$, i.e.,

For any stochastic sequence $\{u(0),u(1),u(2),\cdots\}$, denote its averaged power by $\|u\|_{\cal P}$,