Lecture 23- Stability Analysis for Feedback Control Systems

Lecture 23- Stability Analysis for Feedback Control Systems...

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1 Lecture 23 ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 23: Stability Analysis for Feedback Control Systems Reading: O and W Sections 11.4 and 11.5 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 23 ECSE304 Signals and Systems II Course Outline Discrete-Time Fourier Series and DT Fourier Transform • The Z – Transform • Time and Frequency Analysis of DT Signals and Systems • Sampling Systems • Application to Communications Systems • State Models of Continuous Time LTI Systems – Lecture 18: State Space Analysis – Lecture 19: Solution of State Equations – Lecture 20: Observability and Controllability • Linear Feedback Systems – Lecture 22: Feedback Control Systems – Root Locus Stability – Lecture 23: Stability Analysis – Nyquist Criterion –Lecture 24: Stability Analysis – Gain and Phase Margins
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3 Lecture 23 ECSE304 Signals and Systems II • Observe stability of this system when varying loop gain • Open-loop TF: with poles and zeroes • Close-loop TF: • Stability is determined by closed-loop poles: – System is stable if has no RHP poles – Assumes that has no RHP pole-zero cancellation Review: Root Locus Stability Analysis Controller Plant () et rt ut y t kK s Hs () () KsHs () 1 Ys kKsHs Ts R sk K s H s == + ν μ 1( ) ( ) 0 kK s H s += k ) ( ) kK s H s + 4 Lecture 23 ECSE304 Signals and Systems II • Compute the poles and zeros of Rule 0: The root locus has branches Rule 1: At the root locus starts at the poles of Rule 2: As , - branches starting at poles go to the zeroes of - branches starting at poles go to infinity Rule 3: The root locus is symmetric with respect to the real axis Rule 4: The root locus lies on the real axis to the left of an odd number of singularities (poles and zeroes) Rule 5: The root locus goes to infinity with asymptotes that – Radiate from a centroid located at: – Have angle: Review: Plotting the Root Locus 0 k = k →∞ ( ) poles zeroes νμ ∑∑ ( ) 21 0,1,. .., 1 m m π + = −−
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5 Lecture 23 ECSE304 Signals and Systems II Review: Plotting the Root Locus 6 Lecture 23 ECSE304 Signals and Systems II • Introduction • The Encirclement Property • Nyquist Plots • Review of Bode Plots • The Nyquist Stability Criterion • Gain and Phase Margins Nyquist Stability Analysis
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7 Lecture 23 ECSE304 Signals and Systems II Nyquist Method for Stability Analysis • Goal: To determine if the closed loop system with transfer function T(s) is stable • Recall that, for a particular value of k , T(s) is stable if there are no RHP poles (assuming no pole-zero cancellation in K(s)H(s) ) •T h e Nyquist method allows us to check for the existence of RHP poles directly from the open loop frequency response • We will obtain this result using Nyquist plots and through application of the Encirclement Property () () () 1( ) ( ) kK s H s Ts kK s H s = + K jH j ω 8 Lecture 23 ECSE304 Signals and Systems II The Encirclement Property • Suppose we have an open loop transfer function: … with poles at • Perform a mapping from a finite closed contour C by
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This note was uploaded on 02/08/2012 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.

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Lecture 23- Stability Analysis for Feedback Control Systems...

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