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].
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Cites background from "Event-Triggered Real-Time Schedulin..."
...Compared with continuous event-triggered transmission scheme in [5] and [16], where the special hardware is necessary for successive measurement and computation, the proposed event-triggered scheme in this paper is a discrete event-triggered transmission scheme because it is only dependent on the latest transmitted state {tk h |tk ∈ N} and state error at the discrete sampling sets {jh |j ∈ N}....
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...Compared with those continuous event-triggered schemes in [5] and [16], where the extra hardware is necessary to fulfill the continuous measurements and computation [4], [5], since we only measure the state and compute the error at a constant sampling period, the extra hardware for continuous measurement is not necessary....
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...1) Compared with an open-loop time-triggered periodic communication scheme [2], [10] and a continuous event-triggered scheme [5], [16], the proposed scheme in this paper is a discrete closedloop event-triggered communication scheme; whether or not the measured state should be transmitted is determined by the desired performance, and the error between the state at the current sampling and the state at the latest transmitted sampling....
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Additional excerpts
...ing attention in recent years [121]–[123]....
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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....
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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]....
[...]