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# topic14 - Topic#14 16.31 Feedback Control Systems...

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Unformatted text preview: Topic #14 16.31 Feedback Control Systems State-Space Systems • Full-state Feedback Control • How do we change the poles of the state-space system? • Or, even if we can change the pole locations. • Where do we put the poles? – Heuristics – Linear Quadratic Regulator • How well does this approach work? Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 14–1 Pole Placement Approach • So far we have looked at how to pick K to get the dynamics to have some nice properties ( i.e. stabilize A ) λ i ( A ) λ i ( A − BK ) • Classic Question: where should we put these closed-loop poles? • Approach #1: use time-domain specifications to locate domi- nant poles – roots of: s 2 + 2 ζω n s + ω 2 n = 0 – Then place rest of the poles so they are “much faster” than the dominant behavior. Example: could keep the same damped frequency w d and then move the real part to be 2–3 times faster than real part of dominant poles ζω n Just be careful moving the poles too far to the left because it takes a lot of control effort • Rules of thumb for 2nd order response: 10-90% rise time t r = 1 + 1 . 1 ζ + 1 . 4 ζ 2 ω n Settling time (5%) t s = 3 ζω n Time to peak amplitude t p = π ω n 1 − ζ 2 Peak overshoot M p = e − ζω n t p October 20, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 14–2 • Approach #2: Could also choose closed-loop poles to mimic a system that has performance similar to what you want – Just set pole locations equal to those of the prototype system. – Various options exist • Bessel Polynomial Systems of order k → G p ( s ) = 1 B k ( s ) Figure 1: Bessel root locations. • All scaled to give settling times of 1 second, which you can change to t s by dividing the poles by t s . October 20, 2007 k = 1 2 1.2 1 0.8 0.6 0.4 0.2 Time (sec) Step response of G p for various k Step response G 0.5 1 1.5 2 2.5 3 10 Pole locations of B 1 ( s ) k-4.6200-4.0530 j 2.3400 –-5.0093, - 3.9668 j 3.7845 –-4.0156 j 5.0723, - 5.5281 j 1.6553 – –-6.4480, - 4.1104 j 6.3142 - 5.9268 j 3.0813 – –-4.2169 j 7.5300, - 6.2613 j 4.4018, - 7.1205 j 1.4540 – – –-8.0271, - 4.3361 j 8.7519, - 6.5714 j 5.6786, - 7.6824 j 2.8081 – – –-4.4554 j 9.9715, - 6.8554 j 6.9278, - 8.1682 j 4.1057, - 8.7693 j 1.3616 – – – – 9.6585, - 4.5696 j 11.1838, - 7.1145 j 8.1557, - 8.5962 j 5.3655, - 9.4013 j 2.6655 – – – – j 1.3071 –-4.6835 j- 12.4022, - 7.3609 j 9.3777, - 8.9898 j 6.6057, - 9.9657 j 3.9342, - 10.4278 – – – – 1 2 3 4 5 6 7 8 9 10 Roots of normalized bessel polynomials corresponding to a settling time of 1 second Figure by MIT OpenCourseWare....
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topic14 - Topic#14 16.31 Feedback Control Systems...

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