Sampled-data controller implementation

Question

1. What are the contributions mentioned in the paper "Sampled-data controller implementation" ?

2. What is the way to translate a TDES supervisor into an SD controller?

3. What is the deterministic property of the TDES supervisor?

4. What does z mean when an event is enabled?

Accepted Answer

The setting of this paper is the implementation of timed discrete-event systems ( TDES ) as sampled-data ( SD ) controllers.. In this paper, the authors establish a formal representation of an SD controller as a Moore synchronous finite state machine ( FSM ).. The authors describe how to translate a TDES supervisor to a FSM, as well as necessary properties to be able to do so.. The authors discuss how to construct a single centralized controller as well as a set of modular controllers, and show that they will produce equivalent output.. The authors also discuss a flexible manufacturing system ( FMS ) example and present some FSM translation issues encountered, as well as the FSM version of some of the system ’ s supervisors.

Accepted Answer

To be able to translate a TDES supervisor into an SD controller, the authors only require that it be CS deterministic to ensure that the resulting FSM will be deterministic.

Accepted Answer

Let G = (Y,Σ, δ, yo, Ym) be the TDES plant to be controlled and let S = (X,ΣS, ξ, xo,Xm) be a CS deterministic TDES supervisor for G. Let C = (I, Z,Q,Ω,Φ,qres) be the SD controller for S.

Accepted Answer

When an element is set to 1, this means that corresponding event is enabled and that the controller should make the event occur before the next clock tick, where 0 means it is disabled.

Accepted Answer

The number of state variables for vectors q ∈ Q is defined to be l =∑n j=1 lj , where lj is the number ofstate variables for Qj .

Accepted Answer

As the authors could have multiple controllers, each using their own event orderings, the authors first need define a default bijective map, γg , between their activity event set and a default index set that the authors will use for labeling the events.

Accepted Answer

As discussed in [7], [8], this requires that their supervisor S be SD controllable for their plant G, G have proper time behavior, S-singular prohibitable behavior, that G be complete for S, and that their closed-loop system be ALF and nonblocking.

Accepted Answer

For Σ′ ⊆ Σhib, the authors have ΓZ(Σ′) = [z0, z1, .., zr−1] such that for j = 0, 1, .., r − 1,zj :={1 if (∃σ ∈ Σ′)η(σ) = j 0 otherwiseDefinition 4.6: The authors define for controller C the state set mapping function, Λ : Xsamp → Q, to be an arbitrary injective function of the designer’s choice.

Accepted Answer

For Σ′ ⊆ Σact, the authors have ΓI(Σ′) = [i0, i1, .., iv−1] such that for j = 0, 1, .., v − 1,ij :={1 if (∃σ ∈ Σ′)γ(σ) = j 0 otherwiseDefinition 4.5: Let η be the output event mapping function for controller C.

Accepted Answer

If their supervisor is defined over a subset ΣS ⊂ Σ, the authors would simply add to every state of their supervisor selfloops for each σ ∈ Σ \\ ΣS and use this new supervisor for the translation.

Accepted Answer

|Sn be a CS deterministic supervisor where each modular supervisor Sj = (Xj ,Σj , ξj , xo,j ,Xm,j), j = 1, 2, .., n, is also CS deterministic.

Accepted Answer

Definition 4.2: The input event mapping function for C is defined to be a bijective map γ : ΣS∩Σact → {0, 1, . . . , v− 1} with v = |ΣS ∩ Σact| such that(∀σ1, σ2 ∈ ΣS∩Σact)γg(σ1) < γg(σ2) =⇒ γ(σ1) < γ(σ2)Definition 4.3: The output event mapping function for C is defined to be a bijective map η : ΣS∩Σhib → {0, 1, .., r− 1} with r = |ΣS ∩ Σhib| such that(∀σ1, σ2 ∈ ΣS∩Σhib)γg(σ1) < γg(σ2) =⇒ η(σ1) < η(σ2) As these functions are bijective, their inverse functions always exist.

Accepted Answer

For TDES supervisor S = (X,Σ, ξ, xo,Xm), states reached from the initial state by sampled strings represent states in S that are at least partially observable.

Accepted Answer

Using FSM to define SD controllers is a good choice as it provides a concretedefinition of the controller, yet a FSM still allows a variety of physical implementations such as a programmable logic controller (PLC) [4] program, using digital logic [5], or as a software program on a computer.

Accepted Answer

the authors derive replacement supervisors NewB4Path, NewB6, and NewB7, also shown in Fig. 6.The authors now convert these new supervisors into modular controllers using the method described in Section IV-F.

Accepted Answer

Definition 2.3: For Gi = (Qi,Σi, δi, qo,i, Qm,i) (i = 1, 2), the authors define the synchronous product G1||G2 of the two DES as:(Q1 × Q2,Σ1 ∪ Σ2, δ, (qo,1, qo,2), Qm,1 × Qm,2),where δ((q1, q2), σ) is only defined and equals(q′1, q ′ 2) if σ ∈ (Σ1 ∩ Σ2), δ1(q1, σ) = q ′ 1, δ2(q2, σ) = q ′ 2; (q′1, q2) if σ ∈ Σ1 − Σ2, δ1(q1, σ) = q ′ 1; (q1, q ′ 2) if σ ∈ Σ2 − Σ1, δ2(q2, σ) = q ′ 2.

Accepted Answer

the occurrence operator, Occu : Σ∗ → Pwr(Σ), is defined as:Occu(s) := {σ ∈ Σ | s ∈ Σ∗.σ.Σ∗}Clearly, the authors would have a problem translating a supervisor if two concurrent strings with the same occurrence image are possible at a given sampled state, but they lead to different states in S.

Sampled-data controller implementation

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Sampled-data controller implementation" ?

2. What is the way to translate a TDES supervisor into an SD controller?

3. What is the deterministic property of the TDES supervisor?

4. What does z mean when an event is enabled?

5. What is the number of state variables for Qj?

6. What is the default bijective map for a TDES controller?

7. What does the supervisor need to do to be able to control the plant?

8. what is the state set mapping function for controller C?

9. What is the output event mapping function for controller C?

10. What is the definition of the next-sampling-state function for C?

11. what is the definition of a CS deterministic supervisor?

12. What is the function that maps events to the index value of the variable in the input event mapping?

13. What is the definition of a TDES supervisor?

14. What is the definition of an SD controller?

15. What is the method used to derive supervisors?

16. What is the definition of a DES?

17. What is the definition of occurrence operator?

Figures

Citations

Sampled-data supervisory control

Implementation of Sampled-data Supervisory Control

Implementation of Sampled-Data Supervisory Control

Stability Analysis of Stochastic Hybrid Jump Linear Systems Using a Markov Kernel Approach

References

Supervisory control of a class of discrete event processes

Symbolic Boolean manipulation with ordered binary-decision diagrams

On the supermal controllable sublanguage of a given language

On the supremal controllable sublanguage of a given language

Supervisory control of timed discrete-event systems

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