Process Control: A First Course with MATLAB
Written from the perspective of a student, this text emphasizes the importance of computers in the modern age of teaching and practicing process control.
TABLE OF CONTENTS Process Control: A First Course with MATLAB
Chapter 7: Stability of Closed-loop Systems
When we design a closed-loop system, the specifications may dictate features in dynamic response. However, we cannot do that unless the system is stable. Thus the foremost concern in a control system design is to keep the system stable, which in itself can be used as a design tool.
What Are We Up to?
Analyzing stability of a closed-loop system with three techniques:
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Routh-Hurwitz criterion for the stability region
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Substitution of s = j ? to find roots at marginal stability
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Root-locus plots of the closed-loop poles
7.1. Definition of Stability
Our objective is simple. We want to make sure that the controller settings will not lead to an unstable system. Consider the closed-loop system response that we derived in Section 5.2:
| (7.1) |  |
with the characteristic equation
| (7.2) |  |
The closed-loop system is stable if all the roots of the characteristic polynomial have negative real parts. Or we can say that all the poles of the closed-loop transfer function lie in the left-hand plane (LHP). When we make this statement, the stability of the system is defined entirely on the inherent dynamics of the system and not on the input functions. In other words, the results apply to both servo and regulating problems.
We also see another common definition - bounded-input bounded-output (BIBO) stability: A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such...
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