Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks
Summary (2 min read)
1. Introduction
- Small embedded microprocessors are quickly becoming an essential part of the most diverse applications.
- A particularly interesting example are physically distributed sensor/actuator networks responsible for collecting and processing information, and to react to this information through actuation.
- Common to all these approaches is the underlying principle that better control performance is achieved by providing more CPU time to control tasks.
- Close at the technical level, although addressing very different problems, is the recent work on stabilization under communication constraints [NE00, BL00, EM01, BPZ02, Lib03, NT04].
2. Notation and problem statement
- The authors shall not need the definition1 of ISS in this note but rather the following characterization.
- Definition 2.1. 1See, for example, [Son04] for an introduction to ISS and related notions.
- When ∆ > 0, the control task needs to be executed before the inequality γ(|e|) ≥ σα(|x|) is satisfied in order to account for the delay ∆ between measuring the state and updating the actuators.
- Answering the above questions is the objective of the following sections.
3. Existence of a lower bound for inter-execution times
- The authors start immediately with one of the main contributions of this note.
- Set R is forward invariant for the closed loop system since the execution rule (2.9) guarantees V̇ ≤ 0.
- Theorem 3.1 shows that the simple execution rule (2.9) results in a sequence of inter-execution times for the control task that is guaranteed to be lower bounded provided that ∆ is sufficiently small.
- The techniques used in the proof rely of Lipschitz continuity and are necessarily conservative for general nonlinear systems.
- For linear systems they provide reasonable estimates and one can even provide computable bounds for ∆ as discussed in the next section.the authors.
4. The linear case
- The authors also assume the existence of a linear feedback: u = Kx rendering the closed loop system globally asymptotically stable and where K is a matrix of appropriate dimensions.
- Note that in the linear case any such K renders the closed loop system ISS with respect to measurement errors.
- They are sufficiently accurate to be useful in practical situations as described in the next section.
5. An academic example
- In Figure 1 the authors can see how the error norm never reaches σ|x| even though it goes beyond σ′|x| which is used as execution rule.
- This gap decreases as the state approaches the origin.
- The authors can see that the estimated values, although conservative, do not overestimate the values obtained through simulation by more than a factor of 3.
6. Co-schedulability of stabilizing control tasks
- The authors shall assume a preemptive scheduler in which the control task has the highest priority and thus cannot be preempted by any other task and is executed without delays when γ(|e|) ≥ σα(|x|).
- Note that timing overheads associated with context switching can be captured in the proposed framework by suitably enlarging ∆.
- When a set of control tasks T can be scheduled despite the overhead associated with the control DR task the authors say that T is co-schedulable with the control task.
- Co-schedulability is now ensured by the following sufficient condition where dre denotes the smallest integer greater than r ∈ R. Theorem 6.1.
- Other possibilities are discussed in the next section.
7. Discussion
- 1. Real-time scheduling policies for non-preemptible tasks.
- The simple preemptive scheduling strategy presented in Section 6 relied on the possibility to preempt all but the control task.
- The results presented in this note are also relevant in this more general context since the lower bound on the inter-execution times can be used to construct a timed-automaton model for the control task.
- 2. ISS with respect to actuation errors and networked control systems.
- Similar ideas have been more or less explicitly explored in [YTS02, MA04, NT04].
Did you find this useful? Give us your feedback
Citations
118 citations
117 citations
Cites background from "Event-Triggered Real-Time Schedulin..."
...Traditionally, this aggregated measurement on each sensor will be transmitted at every instant of time k to a remote filter for further processing, whereas, in a practical networked transmission environment, such a consistent transmission strategy may cause excessive consumption of network communication and computational resources, which has been testified in the existing literature on event-triggered transmission mechanisms (see [30]–[37] and many references therein)....
[...]
117 citations
115 citations
Cites background or methods or result from "Event-Triggered Real-Time Schedulin..."
...1 presented in [28], shows how to compute such largest lower bound τλ, which is recalled in Algorithm 1 of Section 6....
[...]
...1 × 10 (similar to the scales seen in [28])....
[...]
...Motivated by the emerging use of economic communications in modern control systems, we address the problem of maintaining system stability in the context of triggering control [28, 22, 30]....
[...]
...4, and which induces a τλ of similar magnitude to those reported in [28]....
[...]
...The ISS-based triggering approach developed in papers [28] and [22] have inspired the derivation of this result....
[...]
115 citations
Cites background from "Event-Triggered Real-Time Schedulin..."
...However, the event-triggering schemes (such as those in [11]–[13]) usually work with continuous measurement of the triggering conditions, which are difficult to be realized in most devices....
[...]
...In view of such potential advantages, various eventtriggering schemes are studied (see [11]–[14])....
[...]
...And the event-triggering schemes (6) could turn into the scheme in [11] when the scalar γi is chosen the same for all modes and Ni is a unit matrix....
[...]
...in [11], different values of scalar γi and matrix Ni are considered....
[...]
References
1,703 citations
"Event-Triggered Real-Time Schedulin..." refers background or methods in this paper
...This idea is an adaptation to the scheduling context of several techniques used to study problems of control under communication constraints [6], [11], [18]....
[...]
...Close at the technical level, although addressing very different problems, is the recent work on stabilization under communication constraints [6], [7], [11], [15], [18], [19]....
[...]
1,533 citations
"Event-Triggered Real-Time Schedulin..." refers background or methods in this paper
...This idea is an adaptation to the scheduling context of several techniques used to study problems of control under communication constraints [6], [11], [18]....
[...]
...Close at the technical level, although addressing very different problems, is the recent work on stabilization under communication constraints [6], [7], [11], [15], [18], [19]....
[...]
1,142 citations
"Event-Triggered Real-Time Schedulin..." refers background in this paper
...1See, for example, [24] for an introduction to ISS and related notions....
[...]
961 citations
799 citations
"Event-Triggered Real-Time Schedulin..." refers background in this paper
...Close at the technical level, although addressing very different problems, is the recent work on stabilization under communication constraints [6], [7], [11], [15], [18], [19]....
[...]