17: Introduction to System Stability- Frequency-Response Criteria

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Frequency-response concepts have been discussed extensively in many previous chapters, most notably in Chapters 4 and 10 relative to methods of calculation and the physical characteristics of frequency response in stable systems. In this chapter, we consider methods using frequency response for assessing the stability of systems.

To examine the stability of a closed-loop control system, we examine in particular the frequency response of the associated open-loop system, for which the transfer function \(O \operatorname{LTF}(s)=G(s) \times H(s)\) was described in Section 16.6. The open-loop system itself is usually manageable, not unstable, even though its closed-loop version might be unstable. Therefore, the frequency-response function \(\operatorname{OLFRF}(\omega) \equiv \operatorname{OLTF}(j \omega)=G(j \omega) \times H(j \omega)\) can typically be measured on actual hardware, even if there is no good mathematical model for the system. The reason for measuring and/or computing \(O L F R F(\omega)\) is that it provides metrics of the absolute and relative stability of the closed-loop system. In other words, to determine the stability of the closed-loop system, it might not be necessary to close the loop with real hardware, possibly risking damage and/or injury.

Recall that root-locus analysis is based on the characteristic equation \(1+G(s) \times H(s)=0\), or \(G(s) \times H(s)=-1\); obviously, the right-hand-side number −1 is important to the closed-loop poles and to closed-loop system stability. Similarly, the number −1 is also highly significant when we evaluate stability using open-loop frequency response \(G(j \omega) \times H(j \omega)\).