S3PR的控制器合併理論之加強

科技部補助專題研究計畫成果報告

期末報告

S3PR 的控制器合併理論之加強

計 畫 類 別 ： 個別型計畫 計 畫 編 號 ： MOST 102-2221-E-004-001- 執 行 期 間 ： 102 年 08 月 01 日至 103 年 10 月 31 日 執 行 單 位 ： 國立政治大學資訊管理學系 計 畫 主 持 人 ： 趙玉 報 告 附 件 ： 出席國際會議研究心得報告及發表論文 處 理 方 式 ： 1.公開資訊：本計畫涉及專利或其他智慧財產權，2 年後可公開查詢 2.「本研究」是否已有嚴重損及公共利益之發現：否 3.「本報告」是否建議提供政府單位施政參考：否

中 華 民 國 104 年 02 月 11 日

中 文 摘 要 ： 最近，創新和有計算效率的方法來設計最大活性結構簡單彈 性製造控制系統已有所報導。但是，目前還不清楚是否反復 的方法，可以實現最小的顯示器配置。早些時候，我們發 現，在α-S3PR（簡單的順序過程與資源的系統。所有依賴虹 吸管是是強烈地）所需的監視器的數量降到最低不能低於基 本虹吸管數量的。這證實了在[]中兩個控制系統例子是一個 最小的顯示器配置，因為他們屬於 α-S3PR 的，他們在每一 個例子中的控制器數量等於基本信標數量。這項工作進一步 探索了一條新的理論，使上述結果可以擴展到非-α-S3PR 的。我們也提出了一個快速的方法來計算無可達性分析的關 鍵活性標記及禁止標記（比上面的方法更快的速度及更少標 記）。在此基礎上，我們提出了一個快速的方法來合併多個 監視器成一個單一的監視器以實現簡單的配置而無須整數線 性規劃（ILP）。然而，就 Ezpeleta 等的例子而言，我們無 法實現如其他方法的 5 個顯示器，。我們將探索背後的理 論，回答下面的問題： 1。理論上和在什麼情況下，我們的合併方法（基於虹吸）不 能達到最低限度的配置嗎？ 2。在這種情況下，借用稍微扭曲的虹吸的方法，我們怎麼實 現最低限度的配置呢？ 這項工作在保持我們在監控控制該領域的領先地位是非常重 要的。 中文關鍵詞： 派翠網，可達狀態圖，彈性製造系統，Petri 子網標記圖 （MG）。

英 文 摘 要 ： Recently, novel and computationally efficient methods to design maximally permissive liveness enforcing supervisors with simple structures for flexible

manufacturing systems have been reported. However, it is unclear whether the iterative approach can achieve the minimal monitor configuration. Earlier we showed that the minimal number of monitors required cannot be less than that of basic siphons in α-S3PR

(systems of simple sequential processes with

resources) with strongly dependent siphons only. This confirms that two of the three controlled systems in [10] are of a minimal monitor configuration since they belong to α-S3PR and their number in each example equals that of basic siphons. This work

further explores a new theory so that the above

results can extend to non- α-S3PR. We also propose a fast way to compute critical live and forbidden

markings (fewer and faster than the above methods) without reachability analysis. Based on that, we proposed a fast method to merge several monitors into a single one to achieve simple configuration without integer linear programming (ILP). However, for the benchmark by Ezpeleta et al., we were unable to achieve 5 monitors as with other approaches. In the ILP approach, when the number of monitors reaches the lower bound, the ILP can be exited to allow a faster solution. We propose to explore the theory behind to answer the following issues:

1. Is it theoretically true and under what condition, our merging method (siphon-based) cannot achieve minimal configuration?

2. In this case, how do we achieve minimal

configuration by twisting the siphon-based approach? This work is important to maintain our leading

position in the field of supervisor control. 英文關鍵詞： Petri nets, reachability graph, FMS, Marked Graph.

行政院國家科學委員會補助專題研究計畫成果報告

S3PR 的器合理之加強

計畫類別：□個別型計畫 □整合型計畫

計畫編號：102-2221-E-004-001-

執行期間： 102 年 8 月 1 日 至 103 年 7 月 30 日

計畫主持人：趙玉

共同主持人：

計畫參與人員：

執行單位：政治大學資管系

中 華 民 國 1 0 3 年 9 月 1 4 日

1

行政院國家科學委員會專題研究計畫成果報告

S3PR 的器合理之加強

計 畫 編 號：NSC 102-2221-E-004-001- 執 行 期 限：102 年 8 月 1 日 至 103 年 7 月 31 日 主 持 人：趙玉 政治大學資管系 共 同 主 持 人： 計畫參與人員：

一、中文摘要

最近，創新和有計算效率的方法來設計最大 活性結構簡單彈性製造控制系統已有所報 導。但是，目前還不清楚是否反復的方法， 可以實現最小的顯示器配置。早些時候，我 們發現，在α-S3PR（簡單的順序過程與資 源的系統。所有依賴虹吸管是是強烈地）所 需的監視器的數量降到最低不能低於基本虹 吸管數量的。這證實了在[]中兩個控制系統例 子是一個最小的顯示器配置，因為他們屬於 α-S3PR 的，他們在每一個例子中的控制器 數量等於基本信標數量。這項工作進一步探 索了一條新的理論，使上述結果可以擴展到 非-α-S3PR 的。我們也提出了一個快速的方 法來計算無可達性分析的關鍵活性標記及禁 止標記（比上面的方法更快的速度及更少標 記）。在此基礎上，我們提出了一個快速的方 法來合併多個監視器成一個單一的監視器以 實現簡單的配置而無須整數線性規劃 （ILP）。然而，就 Ezpeleta 等的例子而言， 我們無法實現如其他方法的 5 個顯示器，。 我們將探索背後的理論，回答下面的問題： 1。理論上和在什麼情況下，我們的合併方法 （基於虹吸）不能達到最低限度的配置嗎？ 2。在這種情況下，借用稍微扭曲的虹吸的方 法，我們怎麼實現最低限度的配置呢？ 這項工作在保持我們在監控控制該領域的領 先地位是非常重要的。 關鍵詞: 派翠網，可達狀態圖，彈性製造系 統，Petri 子網標記圖（MG）。

Abstract

Recently, novel and computationally efficient methods to design maximally permissive liveness enforcing supervisors with simple structures for flexible manufacturing systems have been reported. However, it is unclear whether the iterative approach can achieve the minimal monitor configuration. Earlier we showed that the minimal number of monitors required cannot be less than that of basic siphons in α-S3_{PR (systems of simple sequential}

processes with resources) with strongly dependent siphons only. This confirms that two of the three controlled systems are of a minimal monitor configuration since they belong to α-S3PR and their number in each example equals that of basic siphons. This work further explores a new theory so that the above results can extend to non- α-S3PR. We also propose a fast way to compute critical live and forbidden markings (fewer and faster than the above methods) without reachability analysis. Based on that, we proposed a fast method to merge several monitors into a single one to achieve simple configuration without integer linear programming (ILP). However, for the benchmark by Ezpeleta et al., we were unable to achieve 5 monitors as with other approaches. In the ILP approach, when the number of monitors reaches the lower bound, the ILP can

be exited to allow a faster solution. We propose to explore the theory behind to answer the following issues:

1. Is it theoretically true and under what condition, our merging method (siphon-based) cannot achieve minimal configuration?

2. In this case, how do we achieve minimal configuration by twisting the siphon-based approach?

This work is important to maintain our leading position in the field of supervisor control.

二、緣由與目的

Recent maximally permissive deadlock prevention controls for systems of simple sequential processes with resources (S3_{PR) also} aim at constructing simplest structures in the shortest amount of time for flexible manufacturing systems modeled by Petri nets. The paper in [1] proposes a method to merge several monitors into a single one while not losing the live states. It achieves the same best results in the existing literature while avoiding the time-consuming reachability analysis which does not scale well with the large size of the nets. For a well-known benchmark, the method needs one more monitor than other approaches. Thus, siphon-based merging may not achieve minimal configuration. Although we could reduce one monitor, which monitor to choose to merge seem to be ad-hoc. It is unclear how to select a monitor to reduce for large nets. This paper tackles such an issue successfully.

三、Results

We propose to combine our two new research results: 1. A Control Policy for a Subclass of

Petri Nets without Reachability Analysis [2] and 2. Merging several monitors into one monitor [1]. We a method to find out whether the number of basic monitors can24 be reduced. Recall that once all elementary siphons are controlled so are all dependent siphons and a basic siphon is also an elementary siphon. Also, in \cite{Chao13a}, the number of monitors is lower bounded by that of basic or elementary siphon for a special subclass of S3_{PR, called}

-S$^3$PR. These two facts motivate us to consider monitors for basic siphons to merge the basic siphon S is associated with a number of critical forbidden markings (CFM), where S is unmarked. Each of such CFM can be merged with a CFM of other basic siphons. Indeed, for the well-known benchmark by Ezpeleta et al., we select the monitor for Basic siphon S4 to merge with those of Basic siphons S1, S10, and S16. Three CFMs of S4 are to be merged with those of S1, S10, and S16. Respectively. The corresponding number of monitors required is 5, one less than 6 (number of basic siphons). The result has been published in [3].

四、參考文獻

[1] Gaiyun Liu, Chao, D.Y. and Uzam, Murat, "A Merging Method for the Siphon-Based FMS Maximally Permissive Controllers with Simpler Structures," IMA J Math Control Info, doi: 10.1093/imamci/dnt029, Aug. 2013.

[2] Gaiyun Liu, D. Y. Chao, and Fang Yu, "A Control Policy for a Subclass of Petri Nets without Reachability Analysis," IET Control

Theory & Applications, vol.7, no.8, pp.1131,1141, May 16 2013doi: 10.1049/iet-cta.2012.0426.

[3] Chao, D.Y., 'Improvement on "A Merging Method for the Siphon-Based FMS Maximally Permissive Controllers with Simpler Structures", ' IMA Journal of Mathematical Control and

3 Information, doi:10.1093/imamci/dnu034, July

科技部補助專題研究計畫出席國際學術會議心得報告

日期： 年 月 日

一、 參加會議經過

本報告描述本人出席 2014 11th IEEE International Conference on Networking, Sensing and Control 國際會議報告國際會議過程、感想及建議。2013 年 11 月 16 日本人搭乘中華航空飛 往機洛杉磯至 2014 年 4 月 5 日進行私人行程，2014 年 4 月 6 日搭乘 Delta 航空至邁阿密參 加 ICNSC2014 國際研討會。本人發表報告一篇論文 “Closed Form Formula Construction to Enumerate Control Related States of K-th Order S3PR System (with a Top Left side non-sharing resource place) of Petri Nets”。4 月 9 日會議結束，本人乘 Delta 航空飛往機洛杉磯至 2014 年 4 月 5 日進行私人行程，7 月 26 日搭乘中國國際航空飛機返回臺灣。此次會議內容超廣 泛精彩廣泛主題包括•Bio-informatics, bio-signals and systems、•Collaborative systems

計畫編號

MOST 102－2221－E－004－001－

計畫名稱

S3PR 的控制器合併理論之加強

Enhancement of Theory of Monitor Merging for S3PR

出國人員

姓名

趙玉

服務機構

及職稱

國立政治大學資訊管理學系, 教授

會議時間

2014 年 4 月 7 日

至

2014 年 4 月 9 日

會議地點

SHERATON MIAMI AIRPORT HOTEL & EXECUTIVE MEETING CENTER (3900 NW 21st Street, Miami, FL 33142) Miami, FL, USA

會議名稱

(中文)第 11 屆 IEEE 網絡,感應及控制國際會議

(英文)

11th IEEE International Conference on Networking, Sensing and

Control

發表題目

(中文)建構枚舉 K-階 S3PR 系統控制相關狀態之派翠網的封閉形式公

式（左上側具非共享資源的地方）

(英文)

Closed Form Formula Construction to Enumerate Control Related

States of K-th Order S3PR System (with a Top Left side non-sharing

resource place) of Petri Nets

2

•Complex system management、•Control of networks、•Discrete-event and Hybrid Systems •Distributed intelligent systems、•Emergency mitigation、•Emergency planning、•Emergency

response、•Environmental and ecological systems、•Fuzzy and neural systems •Hazard mitigation、•Heterogeneous wireless networks、•Homeland security •Human factors、•Human adaptive mechatronics、•Human/computer interface •Information systems & infrastructure、•Intelligent vehicle highway systems •Internet of things、•Knowledge based system、•Medical and patient monitoring •Micro/nano, electro-mechanical sensor systems、•Multi-agent systems

•Multi-level multi-objective optimization、•Network security、•Network-based computing systems、•Networked control systems、•Resilient Control systems

•Sensor design, integration and fusion、•Sensor networks、•Smart car and vehicle control •Smart grids、•Smart home environments、•Smart vision and image processing

•Space-based networking、•Tele-robotics、•Urban infrastructure systems •Wireless communications、•Workflow management

本人論文發表於 4 月 7 日星期一，Session title 及議程如下 : Session

Application of Petri Nets to Transportation and Manufacturing

Time: Monday, 07/Apr/2014 3:30 pm - 5:00 pm Location: Yankee

Chair : Yuh Chao (Organzier) Co-chair : Yi-Sheng Huang (Organizer) Full Paper Session

Presentations

Performance Analysis of Scheduling Rules in Remanufacturing Operations Using Stochastic Petri Nets Mi Pan; Weimin Wu

Zhejiang University, China.

In this paper, we propose an approach for modeling a typical remanufacturing shop and analyzing the scheduling rules using the stochastic Petri nets (SPNs). Previous Works primarily concerned on static

scheduling using queuing theory which has difficulty in model extension and complexity analysis, while Petri nets as a well-developed mathematical approach is preferable in considering more complicated stochastic characteristics in scheduling assessment in remanufacturing operations. Based on the proposed SPN

framework, we discuss three basic scheduling rules and present three interdependent scheduling algorithms. Furthermore, the numeric results are provided to illustrate the influence of different parameters and

scheduling rules on the performance.

A Green Wave Band based Method for Urban Arterial Signal Control Bao-lin Ye; Weimin Wu; Xuanhao Zhou; Weijie Mao; Yi-sheng Huang

State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, China.

Optimal signal timing is an efficient and effective method to mitigate traffic congestion in urban road traffic networks. In this paper, we propose a new method for signal-timing optimization of urban arterial roads. The main idea to the method is to design a bi-direction green wave band for arterial roads. In order to reduce delay and stops, an improved arterial road signal coordination approach is developed. In addition, the arterial signal coordination approach has been expanded to deal with the problem of coordination for urban traffic networks coordination control. Finally, simulation experiments are given to illustrate the effectiveness of the proposed method.

Closed Form Formula Construction to Enumerate Control Related States of K-th Order S3PR System (with a Top Left side non-sharing resource place) of Petri Nets

Daniel yuh Chao; Tsung hsien Yu; Sou chein Wu

National Cheng Chi University, Taipei, Taiwan, ROC, Taiwan.

Earlier, Chao pioneered the very first closed-form solution of the number of reachable and other states for marked graphs (MG) and k-th order system which is the simplest class of S3PR (Systems of Simple Sequential Processes with Resources). This paper progresses one step further on enumerating reachable (forbidden, live and deadlock) states for top k-th order systems (one non-sharing resource place in the top position of the left-side process, below denoted as Top-Left ) with a formula depending on parameter k for a subclass of nets with k sharing resources.

Modelling of Traffic Safety Control Systems Using Timed Petri nets Yi-shun Weng; Yi-sheng Huang; Chia-che Ho

Army Academy, Taiwan.

Timed Petri nets (TPNs) are well utilized as a visual and mathematical formalism to model discrete event systems. This paper proposes to use them to model parallel railroad level crossing control systems. Their applications to both single and double-track railroad lines are illustrated. The resulting models allow one to identify and thus avoid critical scenarios in such systems by conditions and events of the model that control the phase of traffic light alternations. Their analysis is performed to demonstrate how the models enforce the phase of traffic transitions by a reachability graph method. Their important properties are verified. This helps advance the state-of-the-art in traffic safety related to the intersection of railroads and roadways.

Petri Net-based Response Policies to Process Module Failure in Time-constrained Single-arm Cluster Tools Yan Qiao; Naiqi Wu; Chunrong Pan; Mengchu Zhou

Guangdong University of Technology, China.

For wafer fabrication, a process module (PM) in cluster tools is prone to failure. It is crucial to deal with such failure in a proper and timely manner. With residency time constraints, if there are feasible periodic schedules in operating a cluster tool before and after a PM failure, it is desired to make it operate continuously when

4

such a failure occurs. A Petri net model is developed to describe the dynamic behavior of a single-arm cluster tool and failure response policies are proposed to deal with a failure. The proposed policies are formulated via simple control laws for their real-time and on-line implementation. An example is presented to show their application.

二、 與會心得

從會議中主要探討之議題，未來可以探討如何以派翠網路應用在相關領域之研究及提出更 好的解決方案。 最近幾年已經看到用戶在他們的應用程序到雲計算環境之間遷移的相關議題越來有越大的 興趣。然而，由於高複雜性，以雲計算為基礎的服務往往經歷及面臨了大量的故障和安全 漏洞問題，導致對用戶的應用程序的可靠性諸多挑戰。目前的可靠性解決方案集中在基礎 設施本身或應用程序的分析，但沒有考慮到系統組件和應用任務之間複雜的相互依存關 係。 這一方面的議題是未來發展必須要面對的，尤其當以雲環境為高度使用及關鍵的核心 系統，因為它必須越來越多強化在關鍵應用上之監視和控制。以用戶為中心，可靠性驅動 的框架，下列幾個方面之議題必須被考慮： 1.簡化及強化在雲基礎設施中使用者部署和維護的應用程序，從而盡量減少暴露在網絡中 的漏洞。這使得用戶能夠以最安全的方式運行他們的雲應用。 2 提供容錯能力，部署自己的雲應用程序的用戶的服務。這種方法允許應用程序從透明的方 式第三方獲得所需的容錯性能，並提高其可靠性和可用性。 為了提供了地球基礎為土壤水分和地表溫度監測和映射基等相關訊息，衛星的被動微波傳 感器已經面世了 30 年。這兩個變量都需要長時間記錄水文和氣象預報等重要資訊以分析氣 候。這些土地的參數呈現具有較大的時間變因，包括強大的晝夜交替、個人衛星覆蓋不同 的時間段、在不同的頻率操作，並且在一天中的不同時間所觀察到的表面資訊等因素。這 些因素必須更依賴資源整合以提供更頻繁的日常觀察，建立較長期的記錄，並更好地了解 晝夜交替。 安全仍是公路運輸領域的主要當務之急。研究和統計顯示，司機的責任為涉及意外的很大 一部分肇因。為防止突發狀況，駕駛輔助系統已經允許修正後的驅動程序，預測系統可能 是可行的解決方案。此外，為了解決增加時間以解決交通的巨額成本，我們的社會所面臨 的流動性有關的新賭注：“自動車”。今天許多新技術使得自動化車輛具可行性：通信， 傳感器，執行器和嵌入式系統正變得越來越強大和可靠的。在解決安全的路徑、高品質的 開放的通信服務，完全自動化的時代是可以被實現的。

三、 發表論文全文或摘要

本人發表論文 “Closed Form Formula Construction to Enumerate Control Related States of K-th Order S3PR System (with a Top Left side non-sharing resource place) of Petri Nets”” 。論

文摘要如下:

Earlier, Chao pioneered the very first closed-form solution of the number of reachable and other states for marked graphs (MG) and k-th order system which is the simplest class of S3PR (Systems of Simple Sequential Processes with Resources). This paper progresses one step further on enumerating reachable (forbidden, live and deadlock) states for top k-th order systems (one non-sharing resource place in the top position of the left-side process, below denoted as Top-Left ) with a formula depending on parameter k for a subclass of nets with k sharing resources.

Daniel Yuh Chao

Tsung Hsien Yu

Sou Chein Wu

Department of Managment Information Systems, National Chengchi University

ICNSC 2014

April 7-9, 2014

Closed Form Formula Construction to

Enumerate Control Related States of K-th

Order S

3

_{PR System (with a Top Left side}

non-sharing resource place) of Petri Nets

Outline



Introduction



Computation of states of k-th order

system forbidden and non-reachable

states



Computation of states of top k-th

order system forbidden and non

reachable states

6

Introduction



Professor

Chao

pioneered the very

first closed-form solution of the

number of reachable and other states

for

marked graphs

(MG) and

k-th

order

S

3

_{PR, Top-Right, Bottom-Right}



This paper progresses one step

further on enumerating reachable

(forbidden, live, and deadlock) states

for

Top-Left

k-th order systems

Introduction



Large PN

models may take

one

month

to complete the reachability

analysis



An alternative

control policy

is

employed, the total number of

reachable states is needed to

estimate the

percentage of lost

states.

Introduction



In

deadlock recovery

, to estimate

deadlock probability among all

reachable states. We needs to

compute the total number of

reachable states to find the

percentage

(deadlock/ reachable

states)

—rather difficult for arbitrary

nets.

Introduction



Why Top-left ?

Different line of thinking and analysis

method between

Top-left, Top-Right

and Bottom-Right

8

Introduction

Top k-th order system :

 A subclass of S3_PR with _k resource places _r 1,

r₂, . . . , r_k shared between two processes N₁ and N₂.

and one non-sharing resource

place

r

’₁

(=

r*

) used by an operation place

p

*

in

_P

1

 M₀(r₁) = M₀(r₂)=. . .M₀(r_k) =

M

₀

(

r’

₁

) = 1.

 N₁(resp. N₂) uses r₁, r₂, …, r_k (resp. r_k,r_k-1, …,r’₁ r₁)

Introduction

Top k-th order system :



M

₀

(

p

0

1

) =

k+1,

M

0

(

p

02

)=

k

, where

p

01

and

p

02

are the idle places in processes

N

₁

and

N

₂



Holder places of

r

_j

in

N

₁

and

N

₂

are

denoted as

p

_j

and

p’

_j

respectively.

 The compound circuit containing r_i, r_i+1, …, r_j-1, r_jis

called (r_i-r_j)-region.

 xy means _r

2 is at x state (x=1,0,-1) and r’1is at

Introduction

Top-Left 3-th order system :

Fig. 5(a) 3-th order top-system. Fig. 5(b) 3-th order top-system reverse Nr.

p’0 t'2 p3 r1 r2 p1 t2 p2 t3 p* t'3 1 t4 r3 p'2 p’3 t’4 N2 N1 t'1 t'2 _p'0 r1 r2 p0 t1 p1 t2 t3 p2 p’2 t'3 p’3 p’1 t'4 1 t4 p3 4 r3 4 1 1 r*_=r’1 p0 t1 1 1 1 r*_=r’1 t'1 p’1 t*3 t*2 p* p’0 N2 N1 1 1 t1 4 4

Computation of states of k-th order system forbidden and non-reachable states

Important Lemmas and Theorems

Lemma 1

: Any forbidden state in N is

nonreachable in N

r

_.

Lemma 2:

Any nonreachable state in

N is a forbidden one or a

10

Computation of states of k-th order system forbidden and non-reachable states

Important Lemmas and Theorems

Theorem 1:

F(k)

=

¥ (k)

–

B(k)

, where

F(k) :

No. of

forbidden

states

¥ (k) :

No. of

nonreachable

states,

B(k) : No. of nonreachable

+empty-siphon

states

Computation of states of k-th order system forbidden and non-reachable states

F(k) ¥ (k)

Computation of states of k-th order system forbidden and non-reachable states

Lemma 4

:

1)

s is a live state if and only if (iff)

s={(y

₁

... y

_k

)| y

_i

=-1 or 0}, or

s={(x

₁

... x

_k

)| x

_i

=1 or 0}.

2)

The set of live states L

_k

={(

x

₁

...x

_k

)

|x

_i

=1 or 0}{(

y

₁

... y

_k

)| y

_i

=-1 or

0}=L

₁



L

₂

.

3)

The total number of

live states

is

2

k

₊₂

k

_-1=2

k+1

_-1.

Computation of states of k-th order system forbidden and non-reachable states

Theorem 2

:

1)

The possible reachable states are

s={( x

₁

x

₂

... x

_j

y

_j+1

... y

_k

)|0



j



k}

={( x

₁

... x

_j

1 y

_j+2

... y

_k

)|1



j



k }



{(y

₁

... y

_k

)}, where x

_i

=1 or 0 (i=1

to j) and y

_p

=0 or -1 (p= j+2 to k).

2)

The total number of

reachable

12

Computation of states of k-th order system forbidden and non-reachable states

Corollary 2

:

1)

The number of

forbidden states

F(k)

=(k-2)2

(k-1)

₊₁

_.

_{(F=Ř – L)}

2)

The number of

nonreachable

states

¥ (k) =

3

k

_{– (k+2)2}

(k-1)

_.

3)

The number of

nonreachable

+empty-siphon

states B(k) =

¥ (k) - F(k) =

3

k

_{– k2}

k

_{– 1.}

Computation of states of k-th order system forbidden and non-reachable states

k R(k) F(k) ¥ (k) B(k) D(k) P(k)(%) 1 3 0 0 0 0 0 2 8 1 1 0 1 12.5 3 20 5 7 2 2 10 4 48 17 33 16 3 6.25 5 112 49 131 82 4 3.57 6 256 129 473 344 5 1.95 7 576 321 1611 1290 6 1.04 8 1280 769 5281 4512 7 0.55 9 2816 1793 16867 15074 8 0.28 10 6144 4097 52905 48808 9 0.15 11 13312 9217 163835 154618 10 0.075 12 28672 20481 502769 482288 11 0.038 13 61184 44801 1533139 1488338 12 0.0196 14 129280 96513 4653689 4557176 13 0.001

Computation of states of Top-Left k-th

order system forbidden and non reachable states

The equivalent

Ne

N

Computation of states of Top-Left k-th

order system forbidden and non reachable states

Lemma 11: Let s=(-1 00 ₀

3 04…0j-1 1j xj+1 xj+2 …xk) be

such that only the top r1-rj siphon in Ner is unmarked.

M is nonreachable in Ne_.

M*=M+r* is reachable in N.

The total number of such M* is 2k-j_.

Theorem 4:The total number of reachable states in N is

14

Computation of states of Top-Left k-th

order system forbidden and non reachable states

Forbidden

markings in Ne may be

live in N (denoted the number of

which as

C(k)

).

Nonreachable

markings in Ne may

be live in N (denoted the number

of which as A(k)).

Computation of states of Top-Left k-th

order system forbidden and non reachable states

Lemma 13: Let s=(1 00 ₀

3 04 … 0j-1 -1j xj+1 xj+2 … xk) correspond to Marking M such that there are unmarked siphons in only the top (r1-rj )-subnet in Ner.

If M(p’j+1)=1 (xj+1=-1), then M’= M+r* is a forbidden

marking (necessarily evolving into unmarked state) in N. M’ may be a live marking in N.

If M(r_j+1)=1(x_j+1=0), then no SMS is unmarked at M’= M+r* in N. M’ may be a live marking in N.

If M(r_j+1)=1(x_j+1=1), then M’= M+r* is a nonreachable state in N.

Computation of states of Top-Left k-th

order system forbidden and non reachable states

Theorem 4:The total number of forbidden markings in N that may be live in Ne _{is C(k)= 2}k-1_-1.

Lemma 15: Let s=(-1 00 ₀

3 04 … 0j-1 1j xj+1 xj+2 … xk) correspond to Marking M in N such that there are unmarked siphons in only the top (r1-rj)-subnet in Ner. Let M’=M+r*.

If M(p’j+1)=1 (xj+1=-1), then M’ is a nonlive marking N.

If M(r_j+1)=1(x_j+1=0), then no SMS is unmarked under M’ in N. M’ is a legal marking in N.

If M(p_j+1)=-1 (x_j+1=1), then M’ is a nonreachable state in N.

The total number of possible live markings at M is 1k-j_.

Computation of states of Top-Left k-th

order system forbidden and non reachable states

Theorem 5:The total number of nonreachable markings in N that may be live in Ne_{is A(k)= k-1.}

Theorem 6:L’(k) =182k-2_+k-4.

Theorem 7: F’(k)= (k-2) 2k_-(k-3).

Theorem 8: ¥’ (k)= 23k_{- (2k+5) 2}(k-1 )₊₁

16

Computation of states of Top-Left k-th

order system forbidden and non reachable states k R’(k) F’(k) ¥ ’(k) L’(k) B’(k) D’(k) P’_(k)(%) 1 6 0 0 0 0 0 2 17 1 1 16 0 1 5.88 3 43 8 11 35 3 2 4.65 4 103 31 59 72 28 3 2.91 5 239 94 247 145 153 4 1.67 6 543 253 915 290 662 5 0.92 7 1215 636 3159 579 2523 6 0.49 8 2687 1531 10435 1156 8904 7 0.26 9 5887 3578 33479 2309 29901 8 0.13 10 12799 8185 105299 4614 97114 9 0.07 11 27647 18424 326647 9223 308223 10 0.036 12 59391 40951 1003491 18440 962540 11 0.0185 13 126975 90102 3061671 36873 2971569 12 0.0095 14 270335 196597 9295603 73738 9099006 13 0.0048

Compare with Top-Right, Bottom-Right

Top-Left Top-right Bottom-Right R’(k) 2R+ 2(k-1)_-1 =(2k+5)2(k-1)_-1 R(k)+3R(k-1)_{=(5k+7) 2}k-2_. (2k+5)2 (k-1)_-1 C(k) 2(k-1)_{-1 k-1 2}(k-1)_-1 A(k) k-1 2k-1_{-1 k-1} L’(k) 182k-2_{+k-4 182}k-2_{+k-4 182}k-2_+k-4 F’(k) (k-2) 2k_{-(k-3) (5k-11) 2}k-2_{-(k-4) (k-2) 2}k_-(k-3) ¥ ’ (n) 23k_-2k+5)2(k-1 )+_{1 23}k_{- (5k+7) 2}k-2 _23k_-(2k+5)2(k-1 )+₁ D’(k) D(k) D(k)+D(k-1) =2k-3 D(k)

Conclusion

A

innovation

research just apply simple

theory to solve complicated problem

To compute in closed form the number

of reachable states of top k-th order

system (a simple version of S3PR)

without constructing a reachability

graph.

Avoid the dire situation of

mid-run

abortion

of reachability analysis due

to exhausted memory

四、 建議

五、 攜回資料名稱及內容

proceedings and one CD.

科技部補助計畫衍生研發成果推廣資料表

日期:2014/09/03

科技部補助計畫

計畫名稱: S3PR的控制器合併理論之加強 計畫主持人: 趙玉 計畫編號: 102-2221-E-004-001- 學門領域: 生產規劃及排程管理

無研發成果推廣資料

102 年度專題研究計畫研究成果彙整表

計畫主持人：趙玉 計畫編號： 102-2221-E-004-001-計畫名稱：S3PR 的控制器合併理論之加強 量化 成果項目 實際已達 成數（被接 受或已發 表） 預期總達成 數(含實際 已達成數) 本計畫 實際貢 獻百分 比 單位 備註（質 化 說 明 ： 如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ...等） 期刊論文 0 0 100% 研 究 報 告 / 技 術 報 告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 6 1 100%

Daniel Yuh Chao, 2014.07, 'Improvement on ’’A Merging Method for the

Siphon-Based FMS Maximally Permissive Controllers with Simpler Structures’’, ' IMA Journal of Mathematical Control and Information, doi:10.1093/imamci/dnu034 impact factor: 0.967 etc 研 究 報 告 / 技 術 報 告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 國外

博士生 1 1 100% 博士後研究員 0 0 100% （外國籍） 專任助理 0 0 100% 其他成果

(

無法以量化表達之 成 果 如 辦 理 學 術 活 動、獲得獎項、重要 國際合作、研究成果 國 際 影 響 力 及 其 他 協 助 產 業 技 術 發 展 之 具 體 效 益 事 項 等，請以文字敘述填 列。) 無 成果項目 量化 名稱或內容性質簡述 測驗工具(含質性與量性) 0 課程/模組 0 電腦及網路系統或工具 0 教材 0 舉辦之活動/競賽 0 研討會/工作坊 0 電子報、網站 0 科 教 處 計 畫 加 填 項 目 計畫成果推廣之參與（閱聽）人數 0

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