This chapter considers the design of event-triggered static output feedback simultaneous H∞ controllers for a collection of networked control systems (NCSs). It is shown that conventional point-to-point wiring delayed static output feedback simultaneous H∞ controllers can be obtained by solving linear matrix inequalities (LMIs) with a linear matrix equality (LME) constraint. Based on an obtained simultaneous H∞ controller, an L2-gain event-triggered transmission policy is proposed for reducing the network usage. An illustrative example is presented to verify the obtained theoretical results.
- networked control systems
- simultaneous stabilization
- static output feedback
- H∞ control.
A networked control system (NCS) is a feedback control system with feedback loop closed through a communication network. As the signal in an NCS is exchanged via a network, the network-induced delay, packet dropout, and limited network bandwidth can degrade the control performance. Many results have been proposed for dealing with these issues [1–5]. In the early stages, the studies on NCSs were mainly based on periodic task models [4–6]. The number of data packets to be transmitted will be large as the sampling period is small. This leads to a conservative usage of network resources and possibly leads to a congested network traffic. Therefore, how to design networked feedback controllers to achieve desired performance with low network usage is an important issue in NCSs.
Recently, some sporadic task models have been presented in NCSs without degrading system performance. An important approach is the event-triggered scheme [7–26]. In , the state transmitting and the control signal updating events were triggered only if the error between the current measured state and the last transmitted state is larger than a threshold condition. In , event-triggered distributed NCSs with transmission delay were studied. Based on the designed event-triggered policy, an allowable upper bound of the transmission delay was derived. In , for distributed control systems, an implementation of event-triggering control policy in sensor-actuator network was introduced. In , the authors concerned with the design of event-triggered state feedback controllers for distributed NCSs with transmission delay and possible packet dropout. Under the proposed triggering policy, the tolerable packet delay and packet dropout were derived. In , an event-triggered control policy was developed for discrete-time control systems. In , under stochastic packet dropouts, an event-triggered control law for NCSs was calculated by the proposed algorithms. In , an event-triggered scheme was developed for uncertain NCSs under packet dropout. In , an event-based controller and a scheduler scheme were proposed for NCSs under limited bandwidth. The NCSs were modeled as discrete-time switched control systems. A sufficient condition for the existences of event-based controllers and schedulers was derived by the LMI optimization approach. Recently, the event-triggered scheme has been extended to control of NCSs for achieving the disturbance attenuation performance [15–21]. In  and , with considering transmission delays, event-triggered state feedback controllers for NCSs were proposed. Criterion for stability and criterion for co-designing both the controller gains and the trigger parameters were derived. In , an event-triggered state feedback control scheme was proposed for guaranteeing finite
All the results in [7–20] are derived in the assumption that the system states are available for measurement. For practical control systems, system states are often unavailable for direct measurement. In the literature, only few results have been proposed for output-based event-triggered NCSs [22–26]. In , a dynamic output feedback event-triggered controller for NCSs was proposed for guaranteeing the asymptotic stability. In  and , by the passivity theory approach, output-based event-triggered policies were derived for guaranteeing the satisfaction of
On the other hand, few results have been proposed in the literature for simultaneous stabilization of NCSs. The consideration of simultaneous stabilization is important since it allows us to design highly reliable controllers that are able to accommodate possible element failures in control systems. As the signal transmitted through network, the solvability of simultaneous stabilization problem of NCSs is quite different to that of point-to-point wiring control systems. Only few results have been proposed for relevant issues [21, 27]. In , based on the average dwell time approach, the simultaneous stabilization for a collection of NCSs was considered. A sufficient condition for guaranteeing simultaneous stabilization was proposed. In , under the assumption that the network communication channel is ideal (no delay, no packet dropout, and no quantization error), we considered the design of state feedback event-triggered simultaneous transmission policies for a collection of NCSs. Under the proposed event-triggered transmission policies, the
It is known that static output feedback controllers are preferred in practical applications since their implementations are much easier than dynamic output feedback controllers. However, the design of static output feedback controllers is much more difficult than dynamic ones. In this chapter, we extend our previous work  to static output feedback case. Furthermore, we consider the network-induced time-varying delay that has not been considered in . We develop an event-triggered static output feedback simultaneous transmission policy for a collection of continuous-time linear NCSs under time-varying delay. It is shown that, under mild assumptions, conventional point-to-point wiring delayed static output feedback simultaneous controllers can be obtained by solving LMIs with a LME constraint. Based on the obtained static output feedback simultaneous controllers, an event-triggered transmission policy was derived for reducing network usage. Different to the results presented in  and  that only considering the design of an event-triggered controller for a single system, this chapter considers the design of a fixed event-triggered controller that is able to
2. Problem formulation and preliminaries
In this section, the problem to be solved is formulated and some preliminaries are given. For simplifying the expressions, we use the same notations
2.1. Problem formulation
Consider a collect of continuous-time control systems:
where is the system state, is the control input, is the controlled output, is the measured output, is the exogenous input, and are constant matrices with appropriate dimensions. Here, for convenience, we assume ,
In this chapter, we consider the case that the feedback loop of system (1) is closed through a real-time network, but not by the conventional point-to-point wiring. Suppose that the sensor node keeps measuring the output signal
If the measured data is not critical for
Our main goal is to design an event-triggered transmission rule to determine whether the currently measured data should be sent to the controller node, such that, under the transmission delay, all possible closed-loop systems in (3) are internally stable and satisfy, for a given constant and for any and ,
Note that, a practical control system may have several different dynamic modes since it may have several different operating points (please see e.g., the ship steering control problem considered in  ). On the other hand, for achieving higher reliability of a practical control system, we may want to design a controller to accommodate possible element failures. With considering possible element failures, a control system can have several different dynamic modes (see e.g., the reliable control problem for active suspension systems considered in ). The problem we considered has a practical importance owing to its high applicability in designing robust and/or reliable controllers.
The following Lemmas will be used later.
For convenience, define .
with . Suppose that . If there exists a function
and a scalar , such that, for all and, along the solutions of (5),
then the system (5) is asymptotically stable. ■
3. Main results
We first consider the design of the event-triggered transmission policy under the assumption that we have a delayed simultaneous controller, and then show how to derive simultaneous controller under transmission delay.
3.1. Event-triggered transmission policy for NCSs under time-varying delay
Define the equivalent time-varying delay
It is clear that
where and . Then, the systems in (3) can be equivalently described as
To derive an event-triggered transmission policy in the presence of transmission delay, assume that, for the systems in (1), we have a conventional delayed static output feedback simultaneous controller:
which is such that all of the possible closed-loop systems in (7) are internally stable and satisfy the condition (4) for . How to get such a delayed static output feedback simultaneous controller will be discussed later.
Define the error signal:
We have the following results.
Along the solutions of the
where and . Then,
From the definition of , it is clear that as . As a result,
Then, by Schur complement and after some manipulations, it can be proved that if (10) holds, we have
That is, under (11),
That is, the
for some constant . In general we set near to 1. ■
3.2. Synthesis of static output feedback delayed simultaneous controllers
In this subsection, we introduce how to derive a conventional delayed simultaneous static output feedback controller (8) such that all of the closed-loop systems (7) are internally stable and satisfy the condition (4). We have the following results.
Then, along the trajectories of the
By Lemma 1 and the Jensen integral inequality , we can show that
As a result,
By noting (17) and the Schur complement, we know that if , where
Moreover, if and only if , where is the matrix obtained by pre- and post-multiplying by :
4. An illustrative example
Suppose that a control system operates at three different operating points. The dynamics at these operating points are different. Suppose that it behaves in the following three possible modes:
We want to design a static output feedback event-triggered controller that is able to
With this controller, by solving (10) we can get solutions:
According to Theorem 1 and Remark 1, the event-triggered policy is (let ):
With the triggering condition (21), the sensor node can determine whether the currently measured data must be transmitted. If the currently measured data is such that condition (21) is violated, it will be discarded for reducing network usage. If the measured data is such that condition (21) holds, it will be sent to the controller node for updating the control signal.
By simulation, for guaranteeing the simultaneous
In this chapter, we develop an event-triggered static output feedback simultaneous transmission policy for NCSs under time-varying transmission delay. With the proposed method, we do not need to switch controllers or event-triggered policies for an NCS with several different operating points. Moreover, the reliability of NCSs can be improved as possible element failures can be accommodated. The implementation of the obtained event-triggered simultaneous controller is easy as it is in the static output feedback framework. One weakness of our result is that the conditions for the existence of static output feedback simultaneous controllers are represented in terms of LMIs with a LME constraint. Standard LMI tools cannot be directly applied to find the solutions. Possible issues for further study include finding less conservative event-triggered transmission policies, considering the possibility of packet dropouts, and relaxing the continuous monitoring requirement at the sensor node by replacing the event-triggered scheme with a self-triggered one.
This work was supported by the National Science Council of the Republic of China under Grant NSC 101-2221-E-019-037.
real vector of dimension
the Euclidean vector norm.
(resp., ) the transpose (resp., inverse) of matrix
(resp., ) the matrix
the symbol * denotes the symmetric terms in a symmetric matrix
the identity matrix of appropriate dimension.
the block diagonal matrix.
min the minimum value of .
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