Elsevier

Information Sciences

Volume 372, 1 December 2016, Pages 579-590
Information Sciences

The minimal signal-to-noise ratio required for stability of control systems over a noisy channel in the presence of packet dropouts

https://doi.org/10.1016/j.ins.2016.08.048Get rights and content

Abstract

The achievable performance of networked control systems (NCSs) are often constrained to the communication channel characteristics. This paper investigates the minimal signal-to-noise ratio (SNR) for stabilization of the single-input single-output (SISO), linear time-invariant systems whose output feedback is subjected to the additive white Gaussian noise (AWGN), and the effect of the packet dropouts which is modeled as a binary stochastic process. The authors considered both cases, with and without the input stochastic disturbance. With the parameterization of all stabilizing controllers, the results showed that signal-to-noise ratio performance is strongly dependent on the plant’s nonminimum phase zeros and unstable poles locations. It was also proven that the packet dropouts and the input stochastic disturbance might degenerate the signal-to-noise ratio performance. Lastly, simulations are provided in order to verify the obtained theoretical analysis results.

Introduction

In recent years, there have been growing interest in the study of feedback control over communication links [3], [5], [10], [30]. A large number of studies related to the control over communications channels including quantization effects, bit-rate limitations, noise, bandwidth constraints and time delays have been obtained [3], [5], [10], [29], [30]. However, there is no general solution for all listed issues, yet researchers are focused on the simplified channel models but each of them highlights just a certain aspects of the overall problem.

Stability and performance of the networked control systems have been widely studied with different assumptions on the communication channels models [6], [11], [13], [22], [23], [24], [25]. The minimal data rate for stabilization of linear system in the case of noiseless, data rate limited channel has been derived in [16], [17]. The signal-to-noise ratio (SNR) is defined as the ratio of channel input power to the noise signal power, and the SNR presents the basic parameter in communication channels characterization, thus it might cause certain limitations on stability and performance of NCSs. In [12], the variations of the parameters formula for singular measure differential systems with impulse effect are given, and the results are used to investigate the stability problem of a boundary input and output systems. The stability in terms of the signal-to-noise ratio has been primarily studied in [2], [18], [19], [20], where the analysis includes the effects of nonminimum phase zero and time delays in the plant. However, it should be noted that in the signal transmission process, packet dropouts occurs unavoidably [14], which will cause the closed-loop systems performance reduction. Furthermore, the frequent packet dropouts might cause higher signal-to-noise ratio required for stabilizing the NCSs. Other significant and meaningful works on consensus performance can be found in [8], [9].

The research presented in this paper is inspired by the results presented in [10], [18], [19]. However, the aforementioned studies are focused on the channel signal-to-noise ratio needed for the stability of NCSs without considering the packet dropouts and noise, and they are all based on the state space method [15], [26]. Other significant and closely related papers can also be found in [7], [27]. It is significant and challenging to study the stability of the NCSs from frequency domain method. A key problem is description of the packet dropouts process in the transfer function form, and then some quantitative and explicit expressions can be given.

Thus, the stability problem, which is expressed by the signal-to-noise ratio (SNR) bound, defined for the imposed power constraint and the additive white Gaussian noise power spectral density, has been investigated in presence of the packet dropouts and stochastic input disturbance. A power constraint on the channel input signal can arise either due to the electronic hardware limitations or the regulatory constraints introduced to minimize the interference to other communication system users. This paper novelties are as follows: firstly, two explicit expressions for the stability of NCSs in terms of the channel signal-to-noise ratio in the presence of the packer dropouts, the AWGN and the stochastic input disturbance have been presented. Secondly, the relationship between the packet dropouts and the channel signal-to-noise ratio have been revealed explicitly, which can be very useful for the design of NCSs.

The paper is organized as follows. In Section 2, the coprime factorization of the transfer function matrices is derived and the Youla parameterization of all stabilizing controllers is introduced, also a brief explanation of all-pass factors of the nonminimum phase transfer function and two lemmas needed in the following analysis are provided. In Section 3, the problem of the channel signal-to-noise ratio over the AWGN feedback channel with packet dropouts is formulated and solved. In Section 4, further investigation of the stochastic input disturbance presence in the feed-forward channel is given. The illustrative example is presented in Section 5. Lastly, the conclusions are summarized in Section 6.

The notations used in this paper are described as follows. z¯ denotes the conjugate of a complex number z. The transpose and conjugate transpose of a vector u and a matrix A are denoted by uT, uH and AT, AH respectively. The open unit disc, the closed unit disc, the unit circle and the complement of D¯ are denoted by D:={zC:|z|<1},D¯:={zC:|z|1},D:={zC:|z|=1}, and D¯c:={zC:|z|>1} respectively. Moreover, ‖ · ‖2 is denoted as the Euclidean vector norm, and ‖ · ‖F is denoted as the Frobenius norm, so GF2:=tr(GHG). The Hilbert space L2 is defined by L2:={G:G(z)measurableinD,G22:=12πππG(ejθ)F2dθ<},with the inner product F,G:=12πππtr(FH(ejθ)G(ejθ))dθ.As it is well known, the L2 admits an orthogonal decomposition into the subspaces H2 and H2 , where H2:={G:G(z)analyticinD¯c,G22:=supr>112πππG(rejθ)F2dθ<},and H2:={G:G(z)analyticinD,G22:=supr<112πππG(rejθ)F2dθ<}.From the above, it follows that for any FH2 and GH2, we would have F,G=0. It should be highlighted that the same notation ‖ · ‖2 will be used to denote these norms, so the meaning of each of these norms will be clear from the context. Let RH denote the set of all stable, proper, rational transfer functions, while the expectation operator is denoted by E{ · }. Finally, if we define cos(u,v):=|uHv|uv. the ∠(u, v) is the principal angle between the two subspaces spanned by u and v.

Section snippets

Preliminaries

In this section, some important factorizations which will be frequently used are introduced. For the rational transfer function (1α)G(z), where α denotes the packet dropouts probability in the communication channel and 0 ≤ α < 1. Its coprime factorization is given by (1α)G(z)=NM,where N,MRH and they satisfy the Bezout identity MXNY=1,for X,Y,RH. Then all the stablizing parameter controllers K can be characterized by the set [4] K:={K:K=(YMQ)(XNQ),QRH}.

For the transfer function G(z),

Channel signal-to-noise ratio in the presence of packet dropouts

The problem which will be discussed in this section is depicted in Fig. 1, wherein G presents the plant model and K presents the controller. In this paper we assume that the system is initially at rest state. The feedback channel refers to the effects of the AWGN and packet dropouts.

In Fig. 1, the signal dr is the binary stochastic process that models the packet dropouts in communication channel, dr(k)={0ifpacketdropoccursattimek,1ifnopacketdropoccursattimek.with the probability distribution

Channel signal-to-noise ratio with input disturbance

The network architecture under consideration is depicted in Fig. 3, wherein input stochastic disturbance exists in the feed-forward channel and the other notations are the same as in the previous section.

If the power spectral density of the input disturbance and the channel noise are denoted as σd2 and σn2, respectively, we can assume that the input disturbance and the channel noise are uncorrelated with each other. By using the same analysis process as in the previous section, we get Pσn2>Sα(z

Illustrative example

The effectiveness of the obtained theoretic results are verified by the example. Consider a discrete SISO plant with transfer function given by G(z)=zkz(z3)(z+0.5),where k ∈ (2, 4). The plant is a right-invertible and has an unstable pole at z=3. Also, it has a nonminimum phase zero at z=k. Based on the coprime factorization, it can be written N(z)=zkz(z+0.2)(z+0.5),M(z)=z3z+0.2,Moreover, based on the allpass factorization, we have L(z)=zk1kz and thus Gm(z)=1kzz(z3)(z+0.5).The different

Conclusions

In this paper, the binary stochastic process is used to model the packet dropouts in communication channel, also the Youla parameterization of two-parameter controllers is employed in order to determine the two lower bounds of the signal-to-noise ratio (SNR) which will guarantee the stability of the networked control systems in the presence of the packet dropouts and the stochastic input disturbance. It was presented that the bounds are tightly dependent on the locations of nonminimum phase

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    This work was partially supported by the National Natural Science Foundation of China under Grants 61472122, 61672245, 61473128, 61370093 and 61373041, and the Postdoctoral Science Foundation of China under Grants 2015T80800 and 2015M582224, and the Province Natural Science Foundation of Hubei under Grant 2014CFB339.

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