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# lec10_print - MAE143b Linear Control Theory and...

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MAE143b Linear Control - Theory and Applications Lecture 10, Thursday Sep. 3, 2:00-4:50pm Prof. R.A. de Callafon [email protected] CONTENTS OF THIS LECTURE State Space and Frequency Design – recap from Lecture 9 – combining State Space and Frequency Domain – example More Feedback Design – model based design – Ziegler-Nichols tuning rules ( sec. 4.4 ) – design by pole placement ( sec. 7.5 ) – root locus graphical design method ( sec. 5.1, 5.3 - 5.5 ) – examples MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 10, Page 1 Frequency Response Design - recap Design of a controller C ( s ) is done by shaping frequency response of loop gain L ( ) = G ( ) C ( ) During ‘loop shaping’ design we use: Knowledge of G ( ) Knowledge of ideal desired shape of L ( ) Construction of C ( ) via asymptotes in Bode plot 10 -2 10 -1 10 0 10 1 10 2 10 -5 10 0 10 5 mag 10 -2 10 -1 10 0 10 1 10 2 -250 -200 -150 -100 w [rad/s] phase Ideal loopgain L ( ) = G ( ) C ( ): LARGE for low frequencies, SMALL for high frequencies. When | L ( ) | ≈ 1, slope of L ( s ) = 1 or -20 dB/dec to obtain phase and gain margin for stability . MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 10, Page 2

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Frequency Response Design - recap Need for phase and gain margin stems from Nyquist stability criterion for a stable loop gain L ( s ): For stable G ( s ) and C ( s ), the closed-loop system is stable if the Nyquist plot of L ( s ) = G ( s ) C ( s ) does not encircle the point -1. Phase Margin (PM) - distance to point -1 measured in phase shift: PM = L ( pm ) + π ω pm is such that | L ( pm ) | = 1 Gain Margin (GM) - distance to point -1 measured in amplitude change: GM = 1 / | L ( gm ) | ω gm is such that L ( gm ) = π -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 real imag PM 1 GM MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 10, Page 3 Frequency Response Design - recap Phase Margin and Gain Margin can be inspected from both the Nyquist (controur) plot or the Bode plot of L ( ) : -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 real imag PM 1 GM Frequency (rad/sec) Phase (deg); Magnitude (dB) Bode Diagrams -20 -10 0 10 20 Gm=3.5218 dB (at 1.4142 rad/sec), Pm=19.567 deg. (at 1.2009 rad/sec) 10 -1 10 0 10 1 -300 -200 -100 0 PM = distance to 180 deg when | L ( ) | = 1 . GM = distance to 0 dB in Bode plot when L ( ) = 180 deg . Matlab commands: nyquist and margin MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 10, Page 4
State Space Design - recap For state space model ˙ x ( t ) = Fx ( t ) + Gu ( t ) y ( t ) = Hx ( t ) perform the following steps: 1. Design a state feedback K R 1 × n for u ( t ) = r ( t ) K x ( t ) such that matrix F G K of closed-loop system ˙ x ( t ) = ( F G K ) x ( t ) + Gr ( t ) y ( t ) = Hx ( t ) has desired eigenvalues (closed-loop poles). 2. Design a observer gain L R n × 1 for state estimator such that matrix F L H of ˙ ˆ x ( t ) = ( F L H x ( t ) + Gu ( t ) + Ly ( t ) ˆ y ( t ) = H ˆ x ( t ) has desired eigenvalues (closed-loop poles). MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 10, Page 5 State Space Design - recap When combining state estimator and state feedback ˙ ˆ x ( t ) = ( F L H x ( t ) + G u ( t ) + L y ( t ) u ( t ) = K ˆ x ( t ) we obtain the state space realization of the (full order) dynamic output feedback controller: C : ˙ ˆ x ( t ) = ( F L H

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lec10_print - MAE143b Linear Control Theory and...

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