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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Citations
78 citations
Cites background from "Event-Triggered Real-Time Schedulin..."
...It was shown in [Tab07] that this time can be obtained from the following equation:...
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78 citations
Cites methods from "Event-Triggered Real-Time Schedulin..."
...The ETC theory is first systematically studied in [28], which is based on the Liapunov stability theory....
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77 citations
Cites background or methods from "Event-Triggered Real-Time Schedulin..."
...Two different event-triggering conditions are studied based on the state error and the control input error, respectively, which are important classes of event-triggering conditions already applied in [5]–[7]....
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...1: Consider the example used in [6], [7] with the plant given by...
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...In [5]–[7], [11] the event-triggering conditions are designed to guarantee a certain decay rate of the Lyapunov functions, among which continuous-time Lyapunov functions are used in [5], [7], [11] and a piecewise discrete-time Lyapunov function is employed in [6]....
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...Event-triggered control has attracted much focus in recent years due to the promising advantage of less resource utilization compared to traditional periodic digital control [1]–[7]....
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77 citations
77 citations
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]....
[...]