Control Theory Applications for Dynamic Production Systems

Control Theory Applications for Dynamic Production Systems
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Control Theory Applications for Dynamic Production Systems Apply the fundamental tools of linear control theory to model, analyze, design, and understand the behavior of dynamic production systems In Control Theory Applications for Dynamic Production Systems: Time and Frequency Methods for Analysis and Design, distinguished manufacturing engineer Dr. Neil A. Duffie delivers a comprehensive explanation of how core concepts of control theorical analysis and design can be applied to production systems. Time-based perspectives on response to turbulence are augmented by frequency-based perspectives, fostering new understanding and guiding design of decision-making. The time delays intrinsic to decision making and decision implementation in production systems are addressed throughout. Readers will discover methods for calculating time response and frequency response, modeling using transfer functions, assessing stability, and design of decision making for closed-loop production systems. The author has included real-world examples emphasizing the different components of production systems and illustrating how practical results can be quickly obtained using straightforward Matlab programs (which can easily be translated to other platforms). Avoiding unnecessary theoretical jargon, this book fosters an in-depth understanding of key tools of control system engineering. It offers: A thorough introduction to core control theoretical concepts of analysis and design of dynamic production systems Comprehensive and integrated explorations of continuous-time and discrete-time models of production systems, employing transfer functions and block diagrams Practical discussions of time response, frequency response, fundamental dynamic behavior, closed-loop production systems, and the design of decision-making In-depth examples of the analysis and design of complex dynamic behavior requiring approaches such as matrices of transfer functions and modeling of multiple sampling rates Perfect for production, manufacturing, industrial, and control system engineers, Control Theory Applications for Dynamic Production Systems will also earn a place in the libraries of students taking advanced courses on industrial system digitalization, dynamics, and design.

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Neil A. Duffie. Control Theory Applications for Dynamic Production Systems

Control Theory Applications for Dynamic Production Systems. Time and Frequency Methods for Analysis and Design

Contents

List of Figures

List of Tables

Guide

Pages

Preface

Acknowledgments

1 Introduction

1.1 Control System Engineering Software

References

Notes

2 Continuous-Time and Discrete-Time Modeling of Production Systems

2.1 Continuous-Time Models of Components of Production Systems

Example 2.1 Continuous-Time Model of a Production Work System with Disturbances

Program 2.1 WIP response calculated using solution of differential equation

Example 2.2 Continuous-Time Model of Backlog Regulation in the Presence of Rush Orders and Canceled Orders

Example 2.3 Continuous-Time Model of Mixture Temperature Regulation using a Heater

2.2 Discrete-Time Models of Components of Production Systems

Example 2.4 Discrete-Time Model of a Production Work System with Disturbances

Program 2.2 WIP Response calculated recursively using difference equation

Example 2.5 Discrete-Time Model of Planned Lead Time Decision-Making

Example 2.6 Exponential Filter for Number of Production Workers to Assign to a Product

2.3 Delay

Example 2.7 Continuous-Time Model of Delay in a Production System

Example 2.8 Discrete-Time Model of Assignment of Production Workers with Delay

2.4 Model Linearization

2.4.1 Linearization Using Taylor Series Expansion – One Independent Variable

Example 2.9 Production System Lead Time when WIP Is Constant and Capacity Is Variable

2.4.2 Linearization Using Taylor Series Expansion – Multiple Independent Variables

Example 2.10 Production System Lead Time when WIP and Capacity are Variable

2.4.3 Piecewise Approximation

Example 2.11 Piecewise Approximation of a Logistic Operating Curve

2.5 Summary

Notes

3 Transfer Functions and Block Diagrams

3.1 Laplace Transform

Example 3.1 Laplace Transform of a Unit Step Function

Example 3.2 Laplace Transform of an Exponential Function of Time

Example 3.3 Laplace Transform of a Decaying Sinusoidal Function

3.2 Properties of the Laplace Transform

3.2.1 Laplace Transform of a Function of Time Multiplied by a Constant

3.2.2 Laplace Transform of the Sum of Two Functions of Time

3.2.3 Laplace Transform of the First Derivative of a Function of Time

3.2.4 Laplace Transform of Higher Derivatives of a Function of Time Function

3.2.5 Laplace Transform of Function with Time Delay

3.3 Continuous-Time Transfer Functions

Example 3.4 Continuous-Time Transfer Function for WIP in a Production System

Example 3.5 Continuous-Time Transfer Function for Order Release Rate Decision-Making

Example 3.6 Continuous-Time Transfer Function for Two-company Production System with Delays

Example 3.7 Transfer Function of First-Order Continuous-Time Mixture Heating with Delay

Program 3.1 Creating a continuous-time transfer function variable with delay

Program 3.2 Alternative method for creating a continuous-time transfer function variable

3.4 Z Transform

Example 3.8 Z Transform of a Unit Step Sequence

Example 3.9 Z Transform of an Exponential Sequence

Example 3.10 Z Transform of a Decaying Sinusoidal Sequence

3.5 Properties of the Z Transform

3.5.1 Z Transform of a Sequence Multiplied by a Constant

3.5.2 Z Transform of the Sum of Two Sequences

3.5.3 Z Transform of Time Delay dT

3.5.4 Z Transform of a Difference Equation

3.6 Discrete-Time Transfer Functions

Example 3.11 Positive and Negative Powers of z

Example 3.12 Discrete-Time Transfer Function Relating WIP to Work Input Rate

Example 3.13 Discrete-Time Transfer Function with Delay Relating Permanent Worker Capacity to Demand

Program 3.3 Creating a discrete-time transfer function variable

Program 3.4 Alternative method for creating a discrete-time transfer function variable

3.7 Block Diagrams

Example 3.14 Block Diagram for Work In Progress in a Production Work System

Example 3.15 Block Diagram for Adjusting Capacity of Cross-Trained and Permanent Workers

3.8 Transfer Function Algebra

3.8.1 Series Relationships

Example 3.16 Discrete-Time Transfer Functions in Series

Program 3.5 Combining discrete-time transfer functions in series

3.8.2 Parallel Relationships

Example 3.17 Continuous-Time Transfer Functions in Parallel

Program 3.6 Combining continuous-time transfer functions in parallel

3.8.3 Closed-Loop Relationships

Example 3.18 Closed-Loop Transfer Function for Continuous-Time Capacity Adjustment

Example 3.19 Closed-Loop Transfer Function for Discrete-Time Control of Actuator Position

Program 3.7 Calculation of closed-loop transfer function

3.8.4 Transfer Functions of Production Systems with Multiple Inputs and Outputs

Example 3.20 Closed-Loop Continuous-Time Transfer Functions for Backlog Regulation

Example 3.21 Closed-Loop Discrete-Time Transfer Functions for Pressing Operation Control

3.8.5 Matrices of Transfer Functions

Example 3.22 Matrix of Continuous-Time Transfer Functions Modeling Warehouse with Two Inputs and One Output

Program 3.8 Creation of continuous-time transfer function matrix

Example 3.23 Matrix of Discrete-Time Transfer Functions Modeling Closed-Loop Control of a Pressing Operation

3.8.6 Factors of Transfer Function Numerator and Denominator

Example 3.24 Factored Continuous-Time Transfer Function

Example 3.25 Factored Discrete-Time Transfer Function

Program 3.9 Factoring a discrete-time transfer function

3.8.7 Canceling Common Factors in a Transfer Function

Example 3.26 Common Factors in a Continuous-Time Transfer Function

Program 3.10 Canceling common factors in a continuous-time transfer function

3.8.8 Padé Approximation of Continuous-Time Delay

Example 3.27 Substitution of Padé Approximation for Continuous-Time Delay

Program 3.11 Padé approximation of continuous-time delay

3.8.9 Absorption of Discrete Time Delay

Example 3.28 Absorption of Delay in a Discrete-Time Transfer Function

Program 3.12 Absorption of delay in a discrete-time transfer function

3.9 Production Systems with Continuous-Time and Discrete-Time Components

3.9.1 Transfer Function of a Zero-Order Hold (ZOH)

3.9.2 Discrete-Time Transfer Function Representing Continuous-Time Components Preceded by a Hold and Followed by a Sampler

Example 3.29 Discrete-Time Model of WIP Reported after Delay

Program 3.13 Discrete-time transfer function representing ZOH, accumulation of work, delay and WIP sampler

Example 3.30 Discrete-Time Model of Mixture Temperature Regulation

Program 3.14 Discrete-time transfer function representing ZOH, amplifier, mixture heating, and temperature measurement

3.10 Potential Problems in Numerical Computations Using Transfer Functions

Example 3.31 Numerical Inaccuracies in Addition of Transfer Functions

Program 3.15 Approaches for summing transfer function variables

3.11 Summary

Notes

4 Fundamental Dynamic Characteristics and Time Response

4.1 Obtaining Fundamental Dynamic Characteristics from Transfer Functions

4.1.1 Characteristic Equation

4.1.2 Fundamental Continuous-Time Dynamic Characteristics

Example 4.1 Effect of Time Constant on Step Response of a First-Order Continuous-Time Production System or Component

Example 4.2 Effect of Damping Ratio and Natural Frequency on Step Response of a Second-Order Continuous-Time Production System or Component

4.1.3 Continuous-Time Stability Criterion

Example 4.3 Relationship Between Proportional Decision-Rule Parameter and Stability of a Continuous-Time First-Order Production System

Example 4.4 Relationship Between Proportional Decision-Rule Parameter and Stability of a Continuous-Time Second-Order Production System

Example 4.5 Stability of a Production System with Continuous-Time Capacity Adjustment with Delay

Program 4.1 Calculation of fundamental dynamic characteristics of production system with continuous-time capacity adjustment and delay

4.1.4 Fundamental Discrete-Time Dynamic Characteristics

Example 4.6 Effect of Time Constant on Step Response of a First-Order Discrete-Time Production System or Component

Example 4.7 Effect of Damping Ratio and Natural Frequency on Step Response of a Second-Order Discrete-Time Production System or Component

4.1.5 Discrete-Time Stability Criterion

Example 4.8 Relationship Between Proportional Decision-Rule Parameter and Stability of a Discrete-Time First-Order Production System

Example 4.9 Stability of Discrete-Time Actuator Position Control

Program 4.2 Calculation of discrete-time dynamic characteristics

4.2 Characteristics of Time Response

4.2.1 Calculation of Time Response

Example 4.10 Response of Continuous-Time Capacity Adjustment to a Unit Step in Work Input

Program 4.3 Calculation of continuous-time step response

Example 4.11 Response of Discrete-Time Capacity Adjustment to a Unit Step in Work Input

Program 4.4 Calculation of discrete-time step response

Example 4.12 Response of Discrete-Time Actuator Position Control to Constant Velocity Command

Program 4.5 Calculation of response to given input function of time

4.2.2 Step Response Characteristics

Example 4.13 Characteristics of Step Response of Continuous-Time Capacity Adjustment

Program 4.6 Calculation of step response characteristics

Example 4.14 Step Response Characteristics of Discrete-Time Temperature Regulation

Program 4.7 Calculation of step response characteristics of discrete-time temperature regulation

Example 4.15 Step Response Characteristics of a System with Two Time Constants

4.3 Summary

Notes

5 Frequency Response

5.1 Frequency Response of Continuous-Time Systems

5.1.1 Frequency Response of Integrating Continuous-Time Production Systems or Components

Example 5.1 Frequency Response of Backlog in a Production System

Program 5.1 Calculation of frequency response of backlog

5.1.2 Frequency Response of 1st-order Continuous-Time Production Systems or Components

Example 5.2 Frequency Response of Continuous-Time Capacity Adjustment

Program 5.2 Calculation of capacity adjustment frequency response

5.1.3 Frequency Response of 2nd-order Continuous-Time Production Systems or Components

Example 5.3 Frequency Response of Mixture Temperature Regulation

Program 5.3 Frequency response of mixture temperature regulation

5.1.4 Frequency Response of Delay in Continuous-Time Production Systems or Components

Example 5.4 Frequency Response of Delay in a 2-Company Production System

Program 5.4 Calculation of frequency response of delay

5.2 Frequency Response of Discrete-Time Systems

5.2.1 Frequency Response of Discrete-Time Integrating Production Systems or Components

Example 5.5 Frequency Response of Warehouse Inventory

Program 5.5 Frequency response of discrete-time warehouse inventory

5.2.2 Frequency Response of Discrete-Time 1st-Order Production Systems or Components

Example 5.6 Discrete-Time Frequency Response of Capacity Provided by Permanent Workers

Program 5.6 Calculation of frequency response of permanent and cross-trained worker capacity adjustments

5.2.3 Aliasing Errors

5.3 Frequency Response Characteristics

5.3.1 Zero-Frequency Magnitude (DC Gain) and Bandwidth

Example 5.7 Zero-Frequency Magnitude and Bandwidth

Program 5.7 Calculation of zero-frequency magnitude and bandwidth

5.3.2 Magnitude (Gain) Margin and Phase Margin

Example 5.8 Open-Loop Magnitude Margin and Phase Margin of Capacity Adjustment

Program 5.8 Calculation of magnitude margin and phase margin

5.4 Summary

Notes

6 Design of Decision-Making for Closed-Loop Production Systems

6.1 Basic Types of Continuous-Time Control

6.1.1 Continuous-Time Proportional Control

6.1.2 Continuous-Time Proportional Plus Derivative Control

6.1.3 Continuous-Time Integral Control

6.1.4 Continuous-Time Proportional Plus Integral Control

6.2 Basic Types of Discrete-Time Control

6.2.1 Discrete-Time Proportional Control

6.2.2 Discrete-Time Proportional Plus Derivative Control

6.2.3 Discrete-Time Integral Control

6.2.4 Discrete-Time Proportional Plus Integral Control

6.3 Control Design Using Time Response

Example 6.1 Time Response Design of Continuous-Time Integral Control of Production Using Metal Forming

Program 6.1 Calculation of settling times and step response for continuous-time integral control of production using metal forming

Example 6.2 Time Response Design of Discrete-Time Proportional Control of Actuator Position

Program 6.2 Proportional discrete-time control of actuator position

Example 6.3 Time Response Design of Discrete-Time Proportional Plus Derivative Control of Actuator Position

Program 6.3 Calculation of settling time for proportional plus derivative discrete-time control of actuator position

6.4 Direct Design of Decision-Making

Example 6.4 Direct Design for Settling Time of Discrete-Time Integral Control of Production Using Metal Forming

Program 6.4 Calculation of parameter Ki for discrete-time integral control of production using metal forming

Example 6.5 Direct Design for Dead-Beat Response of Discrete-Time Integral Control of Production Using Metal Forming

Example 6.6 Direct Design of Damping Ratio and Settling Time for Continuous-Time Proportional Plus Integral Regulation of Mixture Temperature

Example 6.7 Direct Design of Time Constant Cancellation for Continuous-Time Integral-Lead Regulation of Mixture Temperature

6.4.1 Model Simplification by Eliminating Small Time Constants and Delays

Example 6.8 Simplified Design of Closed-loop Production System with WIP and Backlog Regulation

Program 6.5 Calculation of proportional control parameter for backlog regulation using simplified model

6.5 Design Using Frequency Response

Design Guideline 1 High Open-Loop Frequency Response Magnitude at Relatively Low Frequencies

Design Guideline 2 Low Open-Loop Frequency Response Magnitude at Relatively High Frequencies

Design Guideline 3 Relationship Between Open-Loop Phase Margin and Closed-Loop Damping Ratio

Design Guideline 4 Open-Loop Frequency Response Magnitude in the Vicinity of Frequency ωcp

Design Guideline 5 Relationship Between Closed-Loop Bandwidth, Open-Loop Phase Margin, and Frequency ωcp

6.5.1 Using the Frequency Response Guidelines to Design Decision-Making

Example 6.9 Frequency Response Design of Continuous-Time Control of Production Using Metal Forming

Program 6.6 Calculation of continuous-time integral control parameter for production using metal forming

Example 6.10 Frequency Response Design of Proportional Control for Discrete-Time Production System with Delay Using Phase Margin

Program 6.7 Calculate proportional control parameter using phase margin for discrete-time production system with delay

Example 6.11 Frequency Response Design of Discrete-Time Integral-Lead Control for Mixture Heating with Delay

Program 6.8 Calculation of discrete-time integral-lead control for mixture heating with delay

6.6 Closed-Loop Decision-Making Topologies

6.6.1 PID Control

Example 6.12 Equivalent Discrete-Time PID Control

6.6.2 Decision-Making Components in the Feedback Path

Example 6.13 Discrete-Time Lead Control in the Feedback Path

Program 6.9 Calculation of response with lead control in forward and feedback paths

6.6.3 Cascade Control

Example 6.14 Cascade Discrete-Time Control of Force in a Pressing Operation

Program 6.10 Calculate force and position for pressing operation with cascade control

6.6.4 Feedforward Control

Example 6.15 Discrete-Time Regulation of Lead Time With Feedforward Decision-Making

Example 6.16 Discrete-Time Feedforward Control of Force in a Pressing Operation

Program 6.11 Calculation of pressing force with discrete-time feedforward control

6.6.5 Circumventing Time Delay Using a Smith Predictor Topology

Example 6.17 Circumventing Delay in Discrete-Time Regulation of Lead Time Using Work Output Error and a Smith Predictor Topology

Program 6.12 Calculation of response of discrete-time integral-lead control with Smith Predictor topology

6.7 Sensitivity to Parameter Variations

Example 6.18 Sensitivity of Closed-Loop Time Constant to Variation in Production Using Metal Forming Parameter K m

Program 6.13 Calculation of sensitivity of closed-loop time contant to production parameter Km

6.8 Summary

Notes

7 Application Examples

7.1 Potential Impact of Digitalization on Improving Recovery Time in Replanning by Reducing Delays

Program 7.1 Calculation of result of reducing delays in replanning cycle

7.2 Adjustment of Steel Coil Deliveries in a Production Network with Inventory Information Sharing

Program 7.2 Calculation of galvanizing line inventory frequency response

7.3 Effect of Order Flow Information Sharing on the Dynamic Behavior of a Production Network

Program 7.3 Calculation of fundamental dynamic characteristics and time response of a production network without and with order flow information sharing

7.4 Adjustment of Cross-Trained and Permanent Worker Capacity

Program 7.4 Adjustment of permanent and cross-trained worker capacity

7.5 Closed-Loop, Multi-Rate Production System with Different Adjustment Periods for WIP and Backlog Regulation

Program 7.5 Calculation of fundamental dynamic properties and time response for a production system with backlog and WIP Regulation and two different adjustment periods

7.6 Summary

References

Notes

Bibliography

Index

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Neil A. DuffieUniversity of WisconsinMadison, Wisconsin

This book was written for a course entitled Smart Manufacturing at the University of Wisconsin-Madison, taught for graduate students working in industry. It has been heavily influenced by two decades of industry-oriented research, mainly in collaboration with colleagues in Germany, on control theory applications in analysis and design of the dynamic behavior of production systems. Motivated by this experience, the material in this book has been selected to

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Figure 2.9 Response of change in planned lead time to lateness in order completion.

The exponential filter shown in shown in Figure 2.10 is used in a component of a production system to make periodic decisions regarding the workforce that should be assigned to a product when there are fluctuations in demand for the product. The exponential filter has a weighting parameter 0 < α ≤ 1 that determines how significantly the amplitudes of higher-frequency fluctuations in number of workers are reduced with respect to the amplitudes of fluctuations in demand. This reduction is important because making rapid, larger amplitude changes in the number of workers is likely to be costly and logistically difficult.

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