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Unformatted text preview: EE2010 Systems and Control (Chapter 10) 101 ï‚§ We have seen how stability of the closed loop system can be assessed using the root locus approach or by applying the Nyquist stability criterion ï‚§ We have also learnt how the closed loop system is more robust than an open loop system to model uncertainties. This means that if the model of G ( s ) is not accurate, we can still achieve setpoint tracking with the CL system. We call this robust tracking if setpoint tracking can still be achieved even when G ( s ) is inaccurate G ( s ) + ï€ K ( s ) Y ( s ) R ( s ) Robust Stability: Gain and Phase Margins EE2010 Systems and Control (Chapter 10) 102 ï‚§ However, we have not discussed whether the CL loop system will remain stable if our model G ( s ) is not accurate. Robust tracking assumes robust stability, i.e., stability in the face of model inaccuracies. Robust tracking cannot take place if system becomes unstable when G ( s ) is inaccurate ï‚§ How do we quantify robustness? ï‚§ Assumption: G ( s ) is not a precise model of the plant to be controlled. It is impossible to get a precise model in practice. Nevertheless, we use G ( s ) to design K ( s ) for the closed loop control system ï‚§ Question: After K ( s ) has been designed and CL stability has been ensured for the CL system with G ( s ) and K ( s ), will the CL remain stable when K ( s ) is implemented on the real practical system? EE2010 Systems and Control (Chapter 10) 103 ï‚§ Ideal: We want K ( s ) to still stabilize G ( s ) in closed loop even if the model used to design K ( s ) differs significantly from the actual plant ï‚§ Practical reality: Cannot expect K ( s ) to stabilize the CL system for any amount of inaccuracies in G ( s ) ï‚§ Final question: How do we quantify the amount of model inaccuracies that the system can tolerate before the CL system becomes unstable? ï‚§ We study a simple system involving only gain and phase uncertainties Open Loop TF: G ( s ) K ( s ) K ( s ) G ( s ) + ï€ K ( s ) ) ( ) ( 1 ) ( ) ( ) ( s K s G s K s G s G cl ï€« ï€½ G ( s ) CL TF: EE2010 Systems and Control (Chapter 10) 104 ï‚§ Can the CL system tolerate gain uncertainties? ïƒ˜ For example: ïƒ˜ Actual plant has transfer function: ïƒ˜ Model of the plant for control systems design: where K is not known accurately ï‚§ We can model the control system with this uncertain plant model as follows: ï‚§ What is the maximum K before the CL system becomes unstable? ï‚§ Answer to this lies in how large the gain margin of the OL system is G ( s ) + ï€ K Y ( s ) R ( s ) 1 1 ) ( ï€« ï€½ s s G 1 ) ( ï€« ï€½ s K s G EE2010 Systems and Control (Chapter 10) 105 ï‚§ Similarly, can the CL system tolerate phase uncertainties?...
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 Spring '11
 TanKayChen
 Nyquist plot

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