Structures, Signals and Systems

By Martin Braae

This engineering monograph describes a novel mathematical framework for the abstraction and analysis of large-scale interconnected dynamic systems. The proposal has been acknowledged through peer-review as being a contribution to the theory of such systems. Following a brief review of signals and systems, the topic of structures is discussed. This leads to the formulation of the mathematical framework in which the system connections and its dynamics are separated into two independent sets of mathematical equations. The resulting split allows the analysis of system connections alone without recourse the system dynamics. The consequent structure model enables a cost-effective method for determining the structure of industrial plants where the modelling of dynamics can be extremely expensive. The proposed method is demonstrated on a range of examples, first to confirm its predictions and then to illustrate it potential. Lastly the theory is used to determine the significant impact that the deployment of a new sensor had on the structure of a rod-milling plant in the mineral extraction industry.

• Language: English
• Publisher: Martin Braae
• Release date: Sep 9, 2016
• ISBN: 9781370038947

Book preview

MBuct Monograph in

Applied Control Engineering

Structures, Signals and Systems

Martin Braae

Smashwords Edition

Copyright 2016 Martin Braae

First published: 2016

Solely for use in education

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Table of Contents

Preface

Nomenclature

1. Introduction

1.1 Observations

2. Systems and Signals

2.1. Systems

2.2. Signals

2.3. Dynamic Models

2.3.1. Transfer Function Formulations

2.3.2. State Space Formulations

2.3.3. Comment

3. Structures

3.1. The Proposed Theoretical Formulation for Structures

3.1.1. The Mathematical Entities

3.1.2. The Connection Matrices

3.1.3. Connection Stability

3.1.4. The Structure Matrices

3.1.5. An Alternative Derivation of Structure matrices

3.1.6. Comment

3.2. Overall Dynamic Stability

3.3. The Characteristic Structure Matrix

3.3.1. Splitting a Structure into Independent Sub-Structures

3.3.2 Matched Row and Column Swopping

3.4. Summary --- Step-by-Step Procedure

3.4.1 Comment

3.5 Feedback Loops in Structure Models

4. Graded Examples of Connection and Structure Models

4.1 A Process in Unity Feedback with a Controller

4.1.1. Loop Stability

4.1.2. Internal Stability

4.2. A Hypothetical Configuration

4.3 A Two-Degree of Freedom Configuration

4.4 A Non-Separable Configuration of Two Sub-Systems

4.5 A Fully Connected System of Two Sub-Systems

4.6 An Elementary MIMO Configuration with Partial Coupling

4.7 Various Fully Coupled MIMO Configurations

4.7.1. A First Set of Variations

4.7.2. A Second Set of Variations

4.8 MIMO Systems Based on Three Component Systems

4.8.1. A Variation

4.9 Control of a MIMO Process

4.9.1. Pre-Compensation

4.9.2. A First Variation on the MIMO Connections

4.9.3. A Second Variation on the MIMO Connections

4.10. A Problematic Interconnected Dynamic System

4.11. Classification of Structures

4.11.1 A Two-Cluster SISO Structure

4.11.2 A Single-Cluster MIMO Structure

4.11.3 A Dual-Cluster MIMO Structure

4.11.4 A Two-Cluster MIMO Structure

4.12 Observation on Controllability and Observability

4.13 Comment

5. Application to an Industrial Milling Plant

6. A Large Interconnected Dynamic System

6.1. The Plant Structure

6.2. The First Control Configuration

6.3. A Second Control Configuration

6.4. Comment

6.5. A Case from an Industrial Control Project

7. Conclusion

References

Bibliography

Appendices

A. Some Thoughts on Theoretical Frameworks

A.1. Numeric Frameworks

A.2. Frameworks for Dynamic Systems

A.3. Connection and Structure Frameworks

A.4. Observations and Comment

B. Structure Model in State Space Form

C. Connection Model in State Space Form

D. Investigation of an Invalid Structure

D.1. Pragmatic Comment

E. Selected Formulae for a Partitioned Matrix

E.1. Determinant

E.2. Inverse

E.3. Application to the Combined Structure Matrix

About the Author

Acknowledgements

Preface

This monograph describes a novel mathematical framework for formulating and analysing large interconnected dynamic systems like those that arise from the automation of industrial plants.

In the traditions of engineering the proposed theoretical framework is an invention that has been designed to meet a specification rather than an incremental scholarly extrapolation. Its design brief required that:

The abstraction should form an integral mathematical model consisting of two distinct, independent sets of equations that separate the connections of large interconnected dynamic systems from their dynamics. This split should allow the formulation and analysis of connections that define the structure of a system, without recourse to its dynamic models. The abstraction must be based on transfer functions and matrix algebra to hide superfluous detail and to align with a widely applied control engineering method. Finally its dynamic components must accommodate dead-time exactly since pure transport delays are prevalent in industrial plants.

To date the prototype framework has been found to have a number of interesting features relating to the analysis of the structure of interconnected dynamic systems. Primarily its separable model form minimizes the exorbitant cost to control engineering projects of modelling the dynamic components of large industrial plants. This is achieved by focusing on connections that are readily determined by engineering students during vacation work, or by engineers in training. Specifically its ability to make relevant practical deductions from an analysis of process connections alone becomes very attractive for analysing plant structures in industrial-scale control projects.

The project was motivated by the increased interconnection of instrument signals within large industrial operations that the advent of low-cost microprocessor-based hardware of the 1970s enabled. One such connection on a rod milling plant came from a novel particle-size analyser that was installed in the mid-1970s by a progressive gold plant in South Africa. Until that moment control problems within the plant had been handled successfully by SISO control laws in the form of standard PID hardware. Unfortunately the addition of this single instrument for use in a plant-wide control strategy created a MIMO control problem that had not been anticipated.

Such problems were not well understood at the time, even within academic circles where students were encouraged to explore the phenomena of interaction by analogue simulation. For the milling plant the unexpected multivariable control problem delayed the effective deployment of the sophisticated particle-size analyser while various single-variable control strategies were attempted. Eventually a fledgling theory from UMIST that Professor Rosenbrock expounded during a visit to South Africa was investigated by a colleague working on his PhD project to optimize a pilot milling plant. His achievements led to the proposal that a MIMO control strategy be considered for the industrial plant.

The author was fortunate to be involved in the resulting project as a member of the teams of process, instrument and control engineers that were formed by farsighted managers within NIM, CSIR and EDGM to investigate the so-called "mill control problem". The aim of this high-risk project was to improve the performance of a comminution circuit at EDGM using sophisticated control