control with limited communication and message losses☆
Introduction
In this paper, we consider two constraints that arise in remote control systems and formulate a synthesis problem of -type controllers. One is the time sequencing of messages transmitted over a network that connects the plant and the controller. We assume that only one message can be sent by any component sharing the network at a time, and we employ a periodic sequencing scheme for the communication. Such schemes have been commonly used in practice to meet both data rate and real-time requirements when a number of sensors and actuators share a channel.
The other constraint modeled in this work is the random losses of messages during their transmission over the channel. In a remotely controlled system, messages would often be lost or delayed in a random manner due to network congestion or errors. We adopt the strategy such that if the arrival is delayed, messages are regarded as lost. Furthermore, we assume that the information whether a message is lost or not is always communicated as an acknowledgment message to the controller with one-step delay. These messages are implemented in common network protocols such as the TCP. For more on network issues, we refer to [8], [10].
These two constraints have recently been studied in the context of networked control. In [3], time sequencing was first taken into account in a stabilization problem. This problem was further studied in [7], [21]. In [11], in a decentralized control setup, local controllers are allowed to periodically communicate to each other over a network channel. In [14], an control design of a controller using periodic communication (with no loss) was proposed. On the other hand, the effect of message losses on stabilization was studied in [6] for a scalar case. The design of Kalman filters under lossy measurements is dealt with in [19], while an optimal LQG problem is considered in [9]. In [5], a framework for the use of general stochastic channels was introduced. A common point indicated in these works is the existence of a critical loss probability determined by the unstable poles of the plant. Another way to model such systems is via the Markovian jump systems [1], [4], [13]. Issues arising in networked control have provided with new motivations for the research of such systems. In particular, an synthesis method is presented in [18]. See also [20], [22] for related results on random delays.
In the remote control system of this paper, the controller is assumed to be periodic and also to take account of the losses. Consequently, the proposed design can be viewed as a periodic or multirate control problem with random switchings in the system matrices. The main result is a necessary and sufficient condition stated in terms of linear matrix inequalities (LMIs). The design problem can hence be efficiently solved by numerical methods. We note that the results are in particular based on the recent works [17], [18], where a bounded real lemma for Markovian jump linear systems and then a synthesis method for a special case are derived. In this paper, we obtain a generalization to the periodic case; this forms the basis for treatment of systems involving multirate operations as in networked control. Furthermore, we examine the critical values for the loss probabilities mentioned above. It is shown that the unstable plant poles as well as the communication rates contribute to such probabilities.
This paper is organized as follows. In Section 2, the remote control problem under periodic sequencing and random message losses is formulated. Then, in Section 3, we present some preliminary results regarding stability and norm characterizations for periodic systems with random switchings. In Section 4, the main result for the controller design is provided. This is followed by some examples in Section 5, where critical loss probabilities are obtained. We present a numerical example in Section 6 and then concluding remarks in Section 7.
Section snippets
Problem formulation
Consider the remote control system depicted in Fig. 1. The generalized plant G is a discrete-time system and has a state-space equation of the following form:where is the state, is the exogenous input, is the control input, is the controlled output, and is the measurement output. We make the standard assumptions that is controllable and is observable.
The plant is remotely controlled by the
Periodic systems with random switchings
In this section, we characterize the stochastic stability and derive an exact condition for the norm condition.
Consider the system in (4). The first result gives a necessary and sufficient condition for stochastic stability of this system. Proposition 3.1 The system in (4) is stochastically stable if and only if there exists an N-periodic matrix such that andwhere .
This proposition can be proved by following similar lines as in the
control synthesis
In this section, we present the solution to the remote control synthesis problem. In particular, the characterization result in Section 3 is used.
We first give the state-space equation for the closed-loop system :where is the state given by and
Stabilizability using observer-based controllers
In this section, we consider the stabilization of the plant (1) (when ) employing a specific class of controllers with an observer-based structure. We derive some necessary upper bounds on the message loss probabilities and for this case. These bounds are shown to depend on the unstable dynamics of the plant as well as the communication rates in the two channels.
The analysis in this section is motivated by [9], [19]. It is an extension of them for the controller in the form (3), which
Numerical example
In this section, we present a numerical example that illustrates the results of the paper.
We consider the second-order plant as follows: We fix the period of the switchings as . First, we look at the case as in Example 5.1, where there is perfect communication from the controller to the actuator. So the switching pattern for that channel is and the loss probability is . As the other pattern, we let . Thus, messages are
Conclusion
We proposed an design method for a remotely controlled system over a shared network. Two constraints due to the presence of the network have been incorporated: The periodic sequencing of message transmissions and the random message losses. We have developed a bounded real lemma type result for the class of periodic systems with random switchings. This has been used in the design and has led us to an exact condition consisting of LMIs, which can be solved efficiently. Future research will
References (22)
Minimax control of switching systems under sampling
Systems Control Lett.
(1995)- et al.
An LMI approach to periodic discrete-time unbiased filtering
Systems Control Lett.
(2001) - et al.
Stability results for discrete-time linear systems with Markovian jumping parameters
J. Math. Anal. Appl.
(1993) - et al.
Limited communication control
Systems Control Lett.
(1999) - et al.
Optimal control of LTI systems over unreliable communication links
Automatica
(2006) - et al.
Stabilization with control networks
Automatica
(2002) Stabilization of motor networks
Remote stabilization over fading channels
Systems Control Lett.
(2005)- et al.
Feedback control utilizing packet dropping network links
- D. Hristu-Varsakelis, W.S. Levine, (Eds.), Handbook of Networked and Embedded Control Systems, Birkhäuser, Boston,...
Cited by (130)
Design of a Networked Controller for a Two-Wheeled Inverted Pendulum Robot
2019, IFAC-PapersOnLineSurvey on time-delay approach to networked control
2019, Annual Reviews in ControlCitation Excerpt :They can be roughly categorized into the following types based on the resulting closed-loop systems: Switching systems (Ishii, 2008; Zhang & Yu, 2008), asynchronous dynamical systems (Zhang et al., 2001), and jump linear systems with Markov chains (Schenato, 2009; Seiler & Sengupta, 2005). Note that packet dropouts defined in the aforementioned references have two cases, dropped or sent successfully, which are modeled as a Bernoulli or a two-state Markov chain process based on zero-input or hold-input.
Fundamental limitations and intrinsic limits of feedback: An overview in an information age
2019, Annual Reviews in ControlCitation Excerpt :Various extensions to more general classes of systems have been made in the more recent works (Silva, Derpich, & Østergaard, 2010a; Okano & Ishii, 2014, 2017). Control over lossy channels (Elia, 2005; Elia & Eisenbeis, 2011; Feng, Chen, & Gu, 2018a, 2018b; Hespanha et al., 2007; Ishii, 2008; Tsumura, Ishii, & Hoshina, 2009; Quevedo, Silva, & Goodwin, 2008; Schenato et al., 2007; You & Xie, 2010, 2011): Similar bounds can be obtained for control over unreliable channels where data packets drop out randomly. For losses that can be represented as an i.i.d. process, bounds on the loss probability are established in terms of the topological entropy, or more specifically, the reciprocal of the product of the unstable poles.
Quantized stabilization of wireless networked control systems with packet losses
2016, ISA TransactionsMinimax control over unreliable communication channels
2015, AutomaticaIntroduction
2022, Studies in Systems, Decision and Control
- ☆
This work was supported in part by the Ministry of Education, Culture, Sports, Science and Technology, Japan, under Grant no. 17760344.