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1056 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 44, NO. 8, AUGUST 2014 Framework for Optimal Fault-Tolerant Control Synthesis: Maximize Prefault While Minimize Post-Fault Behaviors Qin Wen, Ratnesh Kumar, Fellow, IEEE, and Jing Huang Abstract —In an earlier work, we introduced a framework for fault-tolerant supervisory control of discrete event systems and presented a necessary and sufficient condition for its existence. In this paper, we introduce the synthesis of an optimal fault- tolerant supervisory controller. Given a discrete event plant with both post-fault and prefault behaviors, an optimal fault-tolerant supervisor we synthesize enforces a set of behaviors in which: 1) a recovery is guaranteed within a bounded delay following any fault; 2) all safety and nonblocking properties are satisfied; 3) the enforced set of prefault behaviors is maximized, and 4) a minimal set of post-fault behaviors is tolerated to achieve recovery in a minimal number of steps. An optimal solution requires a simultaneous maximization (of prefault behaviors) and minimization (of post-fault and prerecovery behaviors), which is quite novel. The optimal solution further minimizes the delay of recovery. The computation has complexity quadratic in the size of plant. Index Terms—Convergence, discrete event systems (DESs), fault tolerant control, stability, supervisory control. I. Introduction D ISCRETE event systems (DESs) are systems with dis- crete states that evolve in response to events [24], [25]. Examples include manufacturing systems, communication pro- tocols, reactive software, and asynchronous hardware. A goal of supervisory control [24], [25] of such systems is to enforce a given specification by restricting the behavior of a given system (called plant). The supervisory role is characterized by the fact that at any given plant state, the supervisor determines a set of controllable events to be enabled, so that the plant evolves over enabled events (including the uncontrollable events) without violating a given specification. Systems are subject to failures due to component faults, and fault-tolerance is a property requiring that a system continues to function, perhaps with a degraded performance, even when some of its Manuscript received November 5, 2012; revised May 29, 2013; accepted September 16, 2013. Date of publication December 2, 2013; date of current version July 15, 2014. This work was supported by the National Science Foundation under Grant NSF-ECCS-0801763, Grant NSF-ECCS-0926029, and Grant NSF-CCF-1331390. This paper was recommended by Associate Editor G. Biswas Q. Wen is with Statistics and Control, Inc., West Des Moines, IA 50266 USA (e-mail: [email protected]). R. Kumar and J. Huang are with the Department of Electrical and Com- puter Engineering, Iowa State University, Ames, Iowa 50011 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMC.2013.2291538 components fail. In applications, fault-tolerance is achieved by using redundancy. In our earlier works [1]–[3], we introduced a framework for fault-tolerant supervisory control of discrete event systems. Since a degraded performance may be tolerable after the occurrence of a fault, the post-fault specification is allowed to be more liberal than the prefault one and so it allows a larger set of traces. The condition for the existence of a fault-tolerant controller proposed in [1]–[3] involves the standard notions of controllability, observability and relative-closure, together with the notion of stability [4], which is used to establish bounded delay recovery from a fault. A motivation for our fault-tolerant control framework based on power system operation can be found in [3]. In this paper, we study the synthesis of an optimal fault- tolerant supervisory controller when the required existence conditions (as reported in [1]–[3]) are not satisfied. An optimal fault-tolerant supervisor we synthesize enforces a set of behav- iors in which: 1) a recovery is guaranteed within a bounded delay following any fault; 2) all safety and nonblocking prop- erties are satisfied; 3) the enforced set of prefault behaviors is maximized; and 4) a minimal set of post-fault behaviors is tolerated to achieve recovery in a minimal number of steps. The contributions of the paper include the following. 1) Introduction of the novel notion of optimal fault-tolerant control for discrete-event systems, that maximizes the prefault behaviors while minimizes the post-fault behav- iors, and ensures safety, nonblockingness, and quickest recovery from faults. 2) The control synthesis algorithm presented in the paper is quite novel in that it simultaneously maximizes prefault behaviors as well as minimizes the post-fault behaviors that must be tolerated to assure fastest recovery. This is unique to the DES supervisory control literature (where either a maximization or a minimization is considered, but not both together) and poses its own set of challenges in defining the synthesis problem as well as formulating its solution. 3) Formally establishing that in general the problem has no good solution as the desired property is not preserved under union (even under union of increasing chains). However, under state-feedback control, a unique maxi- mal (i.e., supremal) prefault behaviors can be synthe- sized. Further even in this case, no unique minimal 2168-2216 c 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

EPS Below Grade 105 - XPS Insulation Extracted After Field Exposure Confirms High Water Absorption & Diminished R‐Value

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XPS Insulation Extracted After Field Exposure Confirms High Water Absorption & Diminished R‐Value  

Technical bulletin

EPS Below Grade Series 105 March 2014 

Studies show that as much as 25% of energy loss from a 

structure can be attributed to a lack of insulation. Insulation 

R‐value is directly correlated to maximum energy efficiency 

in a building envelope; higher R‐values translate into 

increased savings.  It is important to understand that in‐situ 

water absorption can diminish the thermal performance of 

building materials and designers must account for this when 

evaluating different insulation choices. 

A test program conducted in August 2008 evaluated the field 

performance of expanded (EPS) and extruded  (XPS) 

polystyrene foam insulation in a side‐by‐side, below grade 

application following a continuous 15‐year installation period.  

When water and R‐value retention were compared 

between EPS and XPS, the results demonstrated that EPS 

insulation outperformed XPS insulation with better  

R‐value retention and a lower moisture absorption.  

In 2013, an independent testing laboratory was again 

commissioned to evaluate the R‐value and water 

absorption from XPS insulation samples extracted from 

several field locations and applications. A summary of the 

2013 test results are shown in Figure 1 and reconfirm 

there is no correlation between the results from 

standardized laboratory test methods and actual field 

exposure for XPS water absorption. Further, the significant 

loss of R‐value associated with XPS water absorption is 

shown. 

In‐situ water absorption for XPS is widely variable 

ranging from 5‐60% by volume. 

The maximum allowable water absorption of <0.3% 

and <0.7% for XPS as specified by ASTM C5781 and 

CAN/ULC‐S7012 are not in the same range as the 

results after in‐situ field exposure. 

R‐value loss for XPS insulation is directly related to the 

percentage of water absorption by volume. 

FIGURE 2 R-Value Retention of XPS After In-Situ Water Absorption

FIGURE 1 In-Situ R-Value Retention & Water Absorption

EPS Below Grade Series 105 Page Two 

Laboratory Versus In‐Situ Test Methodology 

In‐Situ XPS R‐Value Retention 

Diminished When Exposed To 

Moisture 

STANDARDIZED TEST METHODS  

The most common laboratory test methods to 

evaluate moisture absorption call for partial or full 

submersion conditions that do not replicate exposure 

in field applications. Standardized laboratory 

submersion test methods are typically conducted with 

1‐inch thick samples that are submerged for 24 or 96 

hours. These basic laboratory test methods were not 

developed for predicting in‐situ performance, but are 

intended for use in specifications as a means of product 

quality control. 

 

Building insulations are subjected to a wide range of 

conditions and must maintain their performance over 

extended periods of time. In this context the value of 

in‐situ test data is crucial and more important than 

basic laboratory test methods. 

 

IN‐SITU XPS TEST METHODOLOGY  

To investigate the in‐situ performance of extruded 

polystyrene (XPS) insulation, an independent 

laboratory was commissioned to evaluate the field 

performance for various applications being used in 

different climates.    

The XPS samples were excavated under the 

supervision of the independent laboratory to conduct 

R‐value and water absorption measurements. 

Samples from four different locations and 

applications were selected for testing. Two specimens 

were evaluated from each location. The R‐value 

retention and moisture absorption percent for the 

individual samples were analyzed. 

 

 

 

Specimens were tested for thermal resistance using 

ASTM C518 “Standard Test Method for Steady‐State 

Thermal Transmission Properties by Means of the 

Heat Flow Apparatus” immediately after excavation.  

Moisture content was determined by measuring the 

sample weight at the time of removal and again after 

being oven dried. 

 

 

 

 

  

 

Standardized Laboratory Water 

Absorption Test Methods DO 

NOT Predict Long‐Term R‐value 

Performance 

EPS Below Grade Series 105 Page Three 

Roof B 

Wasilla, AK 

Below Grade Insulation  

St. Paul, MN 

Roof A 

Wasilla, AK 

Roadway Insulation 

 Minneapolis, MN 

XPS Moisture & R‐value Retention Percentages 

EPS Below Grade Series 105 Page Four 

BEST PRACTICES It is important to recognize that the successful use of polystyrene foam insulation depends 

upon its correct installation using good building practice. In below grade applications the 

success of the overall system depends, to a large extent, upon provision for adequate drainage 

of water away from the foundation system. The following design considerations should be 

taken into account for below grade applications:  

Direct Water Away From The Foundation  Provide a slope at grade away from the foundation of at least 6” in 10 feet. 

Direct down spouts to drain at least 3 feet away from the building. 

Avoid landscaping that requires excessive watering in the vicinity of the foundation wall. 

Ensure There Is Adequate Subgrade Drainage  Wrap a geotextile filter fabric around the drain tile at the base of the foundation or place 

over the granular fill material over the drain tile. 

Ensure the drain tile has adequate slope to the outflow point. 

Use well‐graded backfill or other appropriate drainage medium to ensure sufficient sub‐

grade drainage adjacent to the foundation wall. 

Use Details To Minimize Leakage Potential  Seal and flash top edge of exterior foundation insulation. 

Embed lower edge of exterior insulation layer at least 6” into perimeter stone over       

drain tile. 

Keep level of perimeter drain tile below the basement floor level. 

Seal tie rod holes and control joints. 

Place vapor barrier on inside face of wall (on the interior of the insulated wall). 

 

CONCLUSION The long‐term application of XPS insulation below grade results in a higher water retention 

and greater loss of R‐value than reported when using standardized laboratory test methods. 

Water absorption results for XPS using ASTM C2723 cannot be correlated to the in‐service 

performance of extruded polystyrene foam insulation.  

 

Material specifications such as ASTM C578 and CAN/ULC‐S701 cannot provide all of the 

answers that a designer may need. These specifications provide physical property 

requirements for the various EPS and XPS insulation types most commonly used. In many 

cases, end use applications require unique physical properties and manufacturers have 

products to meet these needs; designers should consult with manufacturers when a particular 

application requires specific material properties.  

 

REFERENCES 1 ASTM C578 “Standard Specification for Rigid, Cellular Polystyrene Thermal Insulation” 

2 CAN/ULC‐S701 “Standard for Thermal Insulation, Polystyrene, Boards & Pipe Covering” 

3 ASTM C272 “Standard test Method for Water Absorption of Core Materials for Structural Sandwich Materials”  

Copyright ©EPS Industry Alliance 2014

The EPS Industry Alliance publishes technical bulletins to help inform building professionals on the performance characteristics of expanded polystyrene (EPS)

building products. The information contained herein is provided without any express or implied warranty as to its truthfulness or accuracy.

1298 Cronson Blvd., Suite 201 Crofton, MD 21114 (800) 607‐3772 www.epsindustry.org 

Polystyrene Foam Below Grade  

Third‐party research 

published by Oak Ridge 

National Laboratory in April 

2012  further validates the 

findings of Table 1, indicating 

XPS below grade systems can 

experience a 10‐44% loss of 

energy savings performance  

when subjected to moisture 

accumulation in the range of 

8% ‐ 16%.  This study, 

Measurement of Exterior 

Foundation Insulation To 

Assess Durability in Energy 

Saving Performance, 

evaluated six exterior 

insulation systems  spanning 

installation periods of 9 

months to 15 years.  

ADDITIONAL RESEARCH