In this way, operational flexibility is enhanced and the microsystems are designed for a wider range of application

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Design of Reconfigurable Composite Microsystems Based on Hardware/Software Codesign Principles Tianhao Zhang, Krishnendu Chakrabarty, and Richard B. Fair

Abstract—Composite microsystems that integrate mechanical and flu- idic components with electronics are emerging as the next generation of system-on-a-chip. Custom microsystems are expensive, inflexible, and un- suitable for high-volume production. The authors address this problem by leveraging hardware/software codesign principles to design reconfigurable composite microsystems. They partition the system design parameters into nonreconfigurable and reconfigurable categories. In this way, operational flexibility is enhanced and the microsystems are designed for a wider range of application. In addition, the Taguchi robust design method is used to make the system robust, and response surface methodologies are used to explore the widest performance range for the system. A case study is pre- sented for a microvalve, which serves as a representative microelectroflu- idic device.

Index Terms—Application flexibility, nonreconfigurable and reconfig- urable design parameters, response surface, robustness, Taguchi method.

I. INTRODUCTION

Composite microsystems that incorporate microelectromechanical and microelectrofluidic devices are emerging as the next generation of system-on-a-chip (SoC). Composite microsystems combine microstructures with solid-state electronics to integrate multiple coupled energy domains, e.g., electrical, mechanical, thermal, fluidic, and optical, on an SoC. The combination of microelectronics and microstructures enables the miniaturization and integration of new classes of systems that can be used for environmental sensing, control actuation, electromagnetics, biomedical analyses, agent detection, and precision fluid dispensing.

As the number of applications of integrated composite microsystems increases, there is a pressing need for optimization tools to reduce design time, maximize manufacturing yield, and provide high robustness. A number of design methodologies for microsystems has recently been proposed [1], [3], [8]. These methods lead to robust microsystems that meet performance goals and are relatively insensitive to design parameter variations. However, such systems are tailored toward

“custom microsystems” whose performance is designed to be within a narrow range. This leads to expensive and inflexible systems that are not amenable to large-volume production [2].

We propose a reconfigurable composite microsystem design methodology that leverages hardware/software codesign principles to achieve functional unit reusability. The hardware/software codesign method provides design flexibility by allowing software to be compiled efficiently for a modular hardware platform [9]. We show that by partitioning the design parameters of microsystems into two categories—nonreconfigurable “hardware” and reconfigurable

“software” design parameters (referred to as NRDPs and RDPs,

Manuscript received July 31, 2001; revised November 19, 2001. This work was supported in part by the Defense Advanced Research Projects Agency under Contract F30602-98-2-0140. This paper was presented in part in the Proc. Int. Conf. Modeling and Simulation of Microsystems, Hilton Head, SC, USA, pp. 148–152, Mar. 2001. This paper was recommended by Associate Editor R. Gupta.

T. Zhang is with Cadence Design Systems, Inc., Cary, NC 27511 USA (e-mail: tzhang@cadence.com).

K. Chakrabarty and R. B. Fair are with the Department of Electrical and Com- puter Engineering, Duke University, Durham, NC 27708-90291 USA.

Publisher Item Identifier 10.1109/TCAD.2002.800455.

respectively)—we can make the microsystem performance meet the flexibility requirement and be suitable for a wider range of applica- tions. While the values of NRDPs are determined at fabrication time, the values of RDPs are configured (programmed) during operation.

This design approach allows the system to conform to a wider range of performance specification. Such flexible microfluidic components and systems can be used to develop programmable lab-on-a-chip devices as well as electromechanical components that can be produced and sold in high volume. Table I illustrates the partitioning principle for a generic microelectrofluidic system.

The Taguchi experimental design method [11] provides an efficient method for performance variability reduction and is often used for offline parametric optimization control and high-performance design.

The basic idea of this method is to identify the parameters or factors most influential in determining a performance metric and to compute an appropriate setting of the parameters. This is done using orthogonal arrays and design of experiments. We use the Taguchi method to ensure that the system performance lies within an acceptable range and the influence of parametric variations on the system performance is minimized. Statistical response surface analysis studies the system performance variability within a region. Thus, it characterizes the relationships between the basic electrical/mechanical parameters and system performance. This allows a designer to understand how fluctuations in design parameters shift the design point and the associated system behavior and then to explore the maximal system performance range.

The contributions of this paper include the following.

• On the analogy of the hardware/software codesign principles, we partition the set of design parameters into NRDPs and RDPs for application flexibility. This allows the system to be usable for a wider range of application.

• We use the Taguchi experiment design method to determine the values of NRDPs that make the system performance robust, that is, less sensitive to variation of NRDPs.

• We increase the application flexibility of composite microsystems using the response surface methodology.

• Given a range of values that RDPs can take, we design the system such that the range of system performance is maximized under the constraint that the performance is relatively insensitive to the variation of NRDPs.

The organization of the paper is as follows. The general problem statement and design approach are presented in Section II. We describe the Taguchi experiment design method [11], which is used to deter- mine the value of NRDPs for the robust design. We also present the response surface methodology [5] to maximize the performance range for a given set of RDPs. Section III further describes the design pro- cedure for achieving the application flexibility. Section IV presents a case study based on a microvalve, which serves as an example of electrostatic microfluidic devices. Conclusions are presented in Section V.

II. DESIGNAPPROACH

A. NRDPs and RDPs Partitioning

The overall microsystem cost and performance are affected by the partitioning of the design parameters into NRDPs and RDPs. The par- titioning decision is dependent on the relationship between design parameters, system reliability, and cost. Some design parameters can be partitioned explicitly. For example, geometric design parameters, such as the length of the cantilever beam in the microvalve, must be configured before or during fabrication. These are the nonreconfigurable 0278-0070/02$17.00 © 2002 IEEE

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988 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 21, NO. 8, AUGUST 2002

TABLE I

DESIGNPARAMETERS FOR AMICROFLUIDICSYSTEM

Fig. 1. Performance variability reduction.

prefabrication design parameters. Other design parameters can be cat- egorized as either NRDPs or RDPs. The parameters that are config- ured before system execution are classified as NRDPs, while those pa- rameters that can be configured during runtime are classified as RDPs.

These postfabrication parameters can be configured before execution or during field operations. Some factors influencing this partitioning decision are as follows.

1) Correlation

Correlated parameters must be placed in the same category.

These correlations and dependencies are generally determined by the designer. Alternatively, a database can be used to auto- matically extract these correlations. For example, there is significant correlation between the beam width and the perimeter of the moving electrode in accelerometers [12]. Therefore, these two parameters (beam width and perimeter of the electrode) must be placed in the same category.

2) Ease of Control

Some design parameters, e.g., fluid pressure and electrical voltage, are relatively easy to control during operation. There- fore, these can be placed in the RDP set to increase the applica- tion flexibility.

3) Cost

The cost of reconfiguration can also be a driving factor. For example, the channel length in a microvalve is expensive to alter after fabrication. Hence, it is preferably placed in the NRDP set to reduce cost.

The NRDP values are determined at manufacturing time, and this provides a nonreconfigurable “hardware” platform. RDPs constitute the “reprogrammable software” that run on this platform. In this way, composite microsystems provide design flexibility for product evolution and different application purposes.

B. Nonreconfigurable Platform Design Robustness

One of the system optimization objectives is to find NRDP values that make the system performance less sensitive to the fluctuation of

TABLE II ORTHOGONALARRAY

the manufacturing process and the operating environment. The Taguchi experiment design method, which is widely used for offline parametric optimization control and high robust design [11], is used here to achieve this objective.

Fig. 1 illustrates the concept of performance variability (sensitivity) reduction. Consider two design parametersx1 andx2. The relationships between variability in design parametersx1andx2, and the corresponding variability in system performance, are shown in Fig. 1. Due to the nonlinear relationship between the design parameterx1and system performance responsef, the change of design point from x11tox12

results in performance variability reduction. A similar change in design point fromx21tox22yields no performance variability reduction due to the linear relationship between the design parameterx2and system performance responsef.

The aim of performance variability reduction is to identify design parameters that have the most influence on performance variability and then set the values of these parameters to move the design point into the region where the performance sensitivity is minimized. At the same time, design parameters having the least influence on performance variability are used to perform functional tuning to ensure that overall system performance meets target specifications. For example, assume an initial design point of(x11; x22) in Fig. 1. After moving the design parameterx1fromx11tox12to reduce performance variability,

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Fig. 2. Unimodal function for two and three dimensions.

the design parameterx2 can be adjusted from valuex22to valuex21

as a tuning factor to maintain the performance target. In this scenario, design parameterx1 is used to reduce the variability at the expense of nominal system performance. Undesirable shifts in nominal system performance are, in turn, compensated via design parameterx2.

Performance variability is computed based on a statistical metric, signal-to-noise ratio (SNR) [11]. Three common formulations of the signal-to-noise ratio (SNR) objective function are as follows.

• Maximize performance response SNR = 010 log₁₀ 1

n

n i=1

1

y_i² : (1)

• Minimize performance response SNR = 010 log₁₀ 1

n

n i=1

y_i² : (2)

• Target a particular performance specification while minimizing performance variance

SNR = 010 log₁₀ ²

² (3)

where

yi performance response for the ith setting of the parameter combination;

n number of samples of the performance response corresponding to the number of design parameter combinations;

mean of overall performance response, = 1=n ⁿ_i=1yi;

² sample variance of performance response, ² = (1=n 0 1) ⁿ_i=1(yi0 )²:

SNR transforms the performance response into the log domain and provides a standard representation of different design performance variability reduction objectives; the objective functions are generally constructed such that the larger the SNR, the better the performance.

For instance, when the design objective is to maximize a performance metric, the larger the performance response, the larger its associated SNR (1). In a similar manner, when the design objective is to minimize a performance metric, the smaller the performance response, the larger its associatedSNR (2). When SNR is applied to an on-target design, the smaller the performance variance to the target, the larger the associatedSNR (3).

Design parameters are called factors and a particular parameter value is called a level. Combinations of parameters and values (factor levels) are delineated using orthogonal arrays [6]. Orthogonal arrays originate from design of experiments theory for studying a system involving a large number of parameters/variables with a small number of experiments. Parameters are listed horizontally, forming the columns and experiments or combinations of values of the parameters are listed verti- cally, forming the rows. An orthogonal array possesses the property that all columns are mutually orthogonal in that, for any pair of columns, all combinations of factor levels occur and they occur an equal number of times. Table II shows an orthogonal arrayL8 with four columns and four design parameters on two factor levels. These factor levels are denoted by 1 and01, respectively. Performance variability reduction computes the effect of each design parameter at several settings or levels onSNR and uses these results to determine the best combination of parameter settings for optimizing performance stability.

The effect of a design parameter at a particular setting (factor level), called the main effect, is defined as the deviation of the factor level; it is caused from the overall mean of the performance response, and is given by the following equation:

M_x^j = 1n

n i=1

SNRi (4)

where

Mx^j effect of design parameterx1at levelj on SNR;

n number of experiments (simulations) involving the design parameterx1set to levelj.

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For example, the main effect ofx1at levelj = 01 on SNR is computed as the average of theSNRs corresponding to each performance response wherex1is set to levelj = 01. Since the main effect represents how close the performance response caused by a factor level is to the design objective, parameter settings having the largest main effect are desirable. In other words, levels that maximize the SNR result in the minimization of performance variability.

C. Degree of “Programmability” of Reconfigurable Design Parameters

The next design objective is to determine the degree of “programma- bility” of RDPs. Therefore, when RDPs run on the robust nonreconfig- urable “hardware” platform, a composite microsystem can provide the design flexibility for product evolution and different application pur- pose. The following factors must be considered in this context.

1) Microsystem Energy Requirement

The energy supply available for composite microsystems is limited due to miniaturization and integration. Hence, the energy requirement of RDPs must conform to this restriction. For ex- ample, the adjustable range of the electrical voltage must lie in the available voltage range.

2) Physical Implementation

The limitation of physical implementation is also a key for “programmability” of RDPs. For example, the operating frequency of a micropump chamber may be limited by the feasibility of physical implementation.

3) Fabrication Technology and Integration Level

With increasing complexity, the fabrication technology and integrated level also limit the operating range of RDPs.

4) Operational Reliability

Higher degree of “programmability” of RDPs may lead to op- erational reliability problems, and it may be more difficult to maintain accurate control over a wider range.

Therefore, designers should consider the related constraints to determine a rational degree of “programmability” for the reconfigurable design parameters.

D. Determining the Performance Flexibility

Based on a certain setting of NRDPs and the determined programma- bility of RDPs, the composite microsystem performance flexibility can be obtained. The performance range is from the lowest performance to highest performance, and the response surface methodology can be used to identify this performance flexibility.

The response surface methodology can be used to directly represent the geometric relationships between the system performance and design parameters. This helps the designer to understand the causal relationships between how design parameters shift the design point and associated system behavior. Since the scope of variation of RDPs is usually limited, we assume that the relationship between RDPs and the system response can be represented as a unimodal function. This im- plies that on the system response surface, there is exactly one point possessing the minimum performance value and exactly one point possessing the maximum performance value, as shown in Fig. 2. There- fore, the local optimal design point is also the global optimal design point in this design space. While we make the unimodal function assumption here to illustrate our approach, we can handle a system with multimodal response surfaces through piecewise approximation tech- niques. In this case, as well as for the case where the system perfor- mance is not a continuous function of the RDP setting, we can use iter- ative search over subintervals in which the performance is a unimodal and continuous function. We can then compare the RDP setting and performance for each subinterval and choose an appropriate setting.

The minimum and maximum performance points can be formed via iterative search algorithms. When there is just one RDP, the relation- ship between system performance and the RDP can be represented with a curve in theX-Y plane, and a one-dimensional iterative search method, such as Golden section or Fibonacci search method, can be used to find the minimum and maximum performance points [7]. If the number of RDPs is greater than one, the response surface can be used to represent the relationship between system performance and RDPs.

An iterative gradient search method, such as Steepest ascent/descent [5], can be used to find the optimal points.

III. DESIGNMETHOD

The goal of NRDPs/RDPs codesign is to obtain wider system perfor- mance within the feasible programmability range of RDPs and a robust setting of the nonreconfigurable “hardware” platform. Since this optimization problem involves multiple objectives, designers need to trade off each objective to get an appropriate design result. Therefore, the proposed optimization procedure includes six steps.

1) Depending on the partitioning criterion, the design parameters are grouped into RDP and NRDP sets.

2) Select a series of settings of NRDPs as a “hardware” platform.

3) Determine the degree of programmability of RDPs.

4) Using the response surface method and an iterative search algorithm, the minimum and maximum system performance values and related RDP values are found within the determined pro- grammability of RDPs.

5) The system robustness (insensitivity to the variation of NRDPs) is represented usingSNR and optimized using the Taguchi robust design methodology. Since theSNR value for a certain setting of NRDPs also depends on the setting of RDPs within their programmability range, theSNR for this setting of NRDPs may vary with the individual value of RDPs. However, with the unimodal assumption, it is reasonable to estimate the robust- ness for a certain setting of NRDPs using the average ofSNRs which are calculated at the RDP nominal setting value, and the RDP values corresponding to the minimum and maximum per- formance values. The average ofSNR value, SNRⁱ, for theith setting of the NRDPs is given by

SNRⁱ=SNRⁱ+ SNRⁱ+ SNRⁱ

3 ; i = 1; 2; . . . ; n (5) where

SNRⁱ SNR value for the ith setting of the NRDPs with the RDP setting at the minimum performance value;

SNRⁱ SNR value with the nominal setting of RDPs on the ith NRDP setting;

SNRⁱ SNR value with the RDP setting possessing the max- imum performance value on theith NRDP setting;

n total number of the setting of NRDPs.

6) Calculate the main effect for each design parameter at a particular setting. Based on these main effect values, we can obtain the desired performance flexibility and robustness.

IV. MICROVALVEMODELING ANDOPTIMALDESIGNANALYSIS

In this section, we present a case study for an electrostatic microfluidic device, the microvalve. A microvalve behavioral model is developed using the hardware description language, VHDL-AMS. The final optimized design result ensures robustness and a wider performance range for application flexibility.

The pressure-driven check valves are very important to the behavior of the micropump since they determine the flow rate of the micropump.

The major parts of the check valve are a cantilever beam and the valve

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Fig. 3. Schematic view of the opening valve [14].

seat. Normally, the cantilever lies against the valve seat, thus closing the port to fluid flow. In operation, the fluid flow exerts pressure against the cantilever. The cantilever, acting like a spring, deflects and allows the fluid to flow through the valve. The schematic view of the opening valve is shown in Fig. 3 [14].

Our performance parameter here is the static flow rate. It is dependent on the structure parameters and the displacement of the valve, which is determined by the pressure difference

8 =h(x1; x2; . . . ; xn; y) y =f(p)

where8 is the static flow rate, x1; x2; . . . ; xndenote structural parameters, andy is the displacement of valve and p is the pressure difference.

In order to get the analytical behavior of the static fluid flow in the gap between the cantilever and the valve seat, the gap is divided into five pieces (Fig. 3). While studying the relationship between the pressure differencep and the displacement of cantilever beam at the individual regions, the overall analytical result of the flow rate,8, can be treated as a function of pressure differencep and the displacement y [14]

p =

V i=I

1pi(8; y) (i = I; II; III; IV; V ): (6)

In addition, the behavior of the cantilever can be described by a second-order differential equation

my + d _y + ky = pA (7)

wherem is effective mass of the cantilever, including the mass of the cantilever and that of the water surrounding the cantilever. The param- eterd is the damping constant, determinated by the geometry of the cantilever, andk is the spring constant of the cantilever.

By substitutingy = f(p) in (7), we see that the static flow rate is fully determined by the actuated pressure difference and the structural parameters, 8 = h(x1; x2; . . . ; xn; p). VHDL-AMS, as an analog hardware description language, is used to build this nonelectrical model [17], and an analog solver, Saber [16], is used to simulate the microvalve behavioral model

8 = b²s² 0 12b l1

y³ + ls¹³ + 144b² ²

l1

y³ + ls²³

2

+ 2b² p s² (8)

TABLE III NRDPS ANDRDPSSETS

TABLE IV

TOLERANCE FORNONRECONFIGURABLEDESIGNPARAMETERS

TABLE V DESIGNLEVELS FORNRDPS

where i is the different pieces of the gap, which can be I; II; III; IV; V ; is a kinetic energy coefficient relevant to the fluidic velocity profile;h is the height of the valve seat, as shown in Fig. 3,s = y + h; b and l1are the width and the length of the valve seat, respectively;l2 is the length of the cantilever over valve seat;L andb⁰are the length and the width of the cantilever, respectively;h⁰is the thickness of the cantilever; andE is Young’s Modulus.

Depending on the physical principles of the microvalve and the par- titioning criteria, the design parameters can be grouped into the NRDP and RDP sets. The geometric design parameters are grouped into the set of NRDPs, and the pressure differencePais placed in the RDP set.

The partitioning is shown in Table III.

Our design objective is to minimize the variation of the overall flow rate8 due to the fluctuation of design parameters. Here, we assume without loss of generality that the design parameter tolerances are 60.2 m, as shown in Table IV.

To determine the NRDP setting for robust design, we use the three design levels for each NRDP as shown in Table V. Since the fabrica- tion material is silicon, the Young ModulusE is taken to be 146.9 GPa.

In addition, the RDP (pressure differencePa) is assumed to be a sinu- soidal pressure at a frequency of 100 Hz. The amplitude ofPais limited in the range of 5,000 to 15 000 Pa, the nominal design setting value is set to 10 000 Pa.

An exhaustive search to find optimal NRDP setting for robust design is very difficult. The complexity of exhaustive search isO(3ⁿ), where

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Fig. 4. Experiment design.

n is the number of NRDPs. However, most practical systems are domi- nated by some of the design parameters, and most higher order interactions are negligible. Therefore, a1=3^pfraction of the original orthogonal array is used for experimental designs with reducedO(3^(n0p)) complexity, wherep is related to the order of interactions. We use the inner orthogonal arrayL8with two levels (01, 1) for NRDP tolerance and the outer orthogonal arrayL18(Addelman–Kempthorne construc- tion [6]) with three levels (01, 0, 1) for NRDPs setting to directly eval- uate the contribution of individual parameters to overall design robustness [5].

Therefore, by using the one-dimensional iterative Fibonacci search method, the setting points ofPa,Pa(min), and Pa(max), with the minimum flow rate and the maximum flow rate, respectively, can be obtained for each NRDP setting. In addition, by calculating the av- erageSNR value at the Panominal settingPa(nom), Pa(min) and Pa(max), the average robustness of a setting of nonreconfigurable

“hardware” platform is obtained. The design procedure is illustrated in Fig. 4 and is explained as follows.

1) Design the Outer Array

Based on the orthogonal arrayL18, and the three design levels for each NRDP shown in Table V, the outer array is obtained as in the Fig. 4, each row represents a setting of NRDPs. For instance, in the first row, theL value, 01, means that the length of the cantilever is 1280m in this setting.

2) Design the Inner Array

Depending on the NRDP tolerance shown in Table IV and the L8orthogonal array structure [6], we can obtain the inner array for each row of the outer array, meaning each setting of NRDPs.

For example, if the inner array shown in Fig. 4 is developed depending on theith setting of the outer array, the value of L at the first row in the inner array,01, implying that the value of L is 1599.8m.

TABLE VI

AVERAGE RATIO FOR THEDESIGNPARAMETERS

TABLE VII

FLOW-RATERANGE18 [ l min]^{FOR THE}DESIGNPARAMETERS

3) Search the Design Performance

Within the degree of “programming” of the RDP (pressure differencePa), we can obtain three performance values forith NRDP setting by the iterative searching method: minimum flow rate(8ⁱ_min), normal flow rate (8ⁱnom), and the maximum flow rate(8ⁱmax).

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Fig. 5. Plot of design parameter effect on SNR.

Fig. 6. Plot of design parameter effect on flowrate range.

4) Calculate the Robustness for Each Setting of Design Parameters

Based on the SNR objective functions, the related system robustness for three design performances can be calculated as SNRⁱ,SNRⁱ, andSNRⁱ, respectively. The overall robustness of a setting of design performance(SNRⁱ) is the average of eachSNRs, as given in (5).

5) Calculate the Main Effect

As shown in the Table VI, the main effect for design levels of each design parameter is the average of theSNR with the same setting for the whole design solutions. For example, the main effect for the length of the cantileverL at lower level (01), M_L⁰¹= 66:67, is the average of SNR for the design solutions whereL is set to 01.

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Fig. 7. Plot of optimal design points.

TABLE VIII DESIGNINFORMATION

6) Calculate the Flow-Rate for Design Levels

The application flexibility of the system, the range of the flowrate, for each design parameter setting can also be calculated, as shown in Table VII. The18 is the difference between the maximum flowrate and the minimum flowrate within the Performance table in Fig. 4. For example, regardless of other design parameter settings, withL set at the lower level (1280

m), the system flow-rate range 18 is 218.26 l= min Additionally, the following important observations can be made con- cerning the optimal system design.

1) Figs. 5 and 6 illustrate that the microvalve robustness and the flow-rate range for different NRDP setting within the range of RDPs, respectively. The setting of NRDPs is depending on the design objectives (robustness versus performance range); there does not exist a unique design point that satisfies conflicting design requirements.

2) In studying robust design, we note that the length of the cantilever(L), the width of the cantilever (b⁰), the thickness of the cantilever(h⁰), and the width of the valve seat (b) have a significant effect onSNR. Except for h⁰, they also have a significant effect on the average flow-rate range. The setting withL(01), b⁰₍₊₁₎,h⁰₍₊₁₎,h₍₀₎,l_{1 (0)},b₍₊₁₎,l_{2 (01)}is clearly the most robust.

The robustness of this setting of the design parameters,SNR, is the average of the main effects for each design parameter.

3) In attempting design for wider flow-rate range design, we note that the length of the cantilever(L), the width of the cantilever

(b⁰), and the width of the valve seat (b) have a significant effect on the average flow-rate range. The setting withL₍₊₁₎, b₍₊₁₎⁰ ,h⁰₍₀₁₎,h(+1),l1 (01),b(+1),l2 (+1)possesses the widest flow-rate range.

Fig. 7 and Table VIII present the optimal design results: optimal design for the widest flow-rate range and the optimal design for robustness. In addition, Figs. 5 and 6 also directly provide very useful infor- mation for related performance improvement. For example, increasing the value ofL, b⁰, and b increases the range of flow rate, while de- creasing the value ofL and increasing the value of b improves the microvalve robustness. Based on these performance analyzes, other feasible design solutions can also be obtained.

V. CONCLUSION

We have leveraged hardware/software codesign principles for the design of reconfigurable composite microsystems. Operational flexibility and system robustness are enhanced by partitioning the design parameters into nonrecofigurable and reconfigurable parameters and through the use of the Taguchi experiment design method. A case study for a microvalve demonstrates the flexibility of this approach.

ACKNOWLEDGMENT

The authors would like to acknowledge the contributions of F. Cao in developing the microvalve simulation model.

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