ME451: Control Systems. Course roadmap

Size: px

Start display at page:

Download "ME451: Control Systems. Course roadmap"

Charlotte Randall
5 years ago
Views:

1 ME451: Control Systems Lecture 20 Root locus: Lead compensator design Dr. Jongeun Choi Department of Mechanical Engineering Michigan State University Fall Modeling Course roadmap Analysis Design Laplace transform Transfer function Models for systems electrical mechanical electromechanical Block diagrams Linearization Time response Transient Steady state Frequency response Bode plot Stability Routh-Hurwitz Nyquist Design specs Root locus Frequency domain PID & Lead-lag Design examples (Matlab simulations &) laboratories Fall

2 Closed-loop loop design by root locus Designable! C(s) Controller G(s) Plant Fixed! Place closed-loop loop poles at desired location by tuning the gain C(s)=K. If root locus does not pass the desired location, then reshape the root locus by adding poles/zeros to C(s). (How?) Compensation (for time domain specs) Fall General effect of addition of poles Pulling root locus to the RIGHT Less stable Slow down the settling Add a pole Add a pole Fall

3 General effect of addition of zeros Pulling root locus to the LEFT More stable Speed up the settling Add a zero Fall Adding only zero Some remarks often problematic because such controller amplifies the high-frequency noise. Adding only pole often problematic because such controller generates a less stable system (by moving the closed-loop loop poles to the right). These facts can be explained by using frequency response analysis. Add both zero and pole! Fall

4 Lead and lag compensators C(s) Controller G(s) Plant Lead compensator Lag compensator Why these are called lead and lag? We will see that from frequency response in this class. Fall Lead compensator Positive angle contribution Test point s -p 1 -z 1 Fall

5 Lag compensator Negative angle contribution Test point s -z 2 -p 2 Fall Roles of lead and lag compensators Lead compensator (Today) prove transient response prove stability Lag compensator (Next) duce steady state error Lead-lag compensator (Next) Take into account all the above issues. Fall

6 Radar tracking system Fall Lead compensator design Consider a system C(s) Controller G(s) Plant Analysis of CL system for C(s)=1 Damping ratio ζ=0.5 Undamped natural freq. ωn=2 rad/s Performance specification Damping ratio ζ=0.5 Undamped natural freq. ωn=4 rad/s Desired pole CL pole with C(s)=1 Fall

7 Angle and magnitude conditions (review) A point s to be on root locus it satisfies Angle condition Odd number For a point on root locus, gain K is obtained by Magnitude condition Fall Lead compensator design (cont d) Evaluate G(s) ) at the desired pole. o o If angle condition is satisfied, compute the corresponding K. In this example, Desired pole Angle condition is not satisfied. Angle deficiency Fall

8 Lead compensator design (cont d) To compensate angle deficiency, design a lead compensator C(s) satisfying Desired pole There are many ways to design such C(s)! Fall Lead compensator Positive angle contribution Test point s Triangle -p 1 -z 1 Fall

9 How to select pole and zero? Draw horizontal line PA Draw line PO Draw bisector PB A Desired pole P Draw PC and PD -p(= p(=-5.4) -z(= z(=-2.9) Pole and zero of C(s) ) are shown in the figure. C B D O Fall Comparison of root locus G(s) G(s)C(s) proved stability! Fall

10 How to design the gain K? Lead compensator Open loop transfer function Magnitude condition Fall Comparison of step responses Compensated system 1.4 Uncompensated system (C(s( C(s)=1) Lead compensator gives faster transient response (shorter rise and settling time) improved stability Fall

11 Error constants Step-error error constant 5 4 Unit ramp input Ramp-error constant Ramp response Lag compensator can be used to reduce steady-state state error. (Next lecture) Fall Summary and exercises Controller design based on root locus General effects of addition of pole and zero Lead lag compensator realization with op amp Lead compensator design Lead compensator improves stability and transient response. Next, lag & lead-lag lag compensator design Fall

ECE317 : Feedback and Control

ECE317 : Feedback and Control Lecture : Frequency domain specifications Frequency response shaping (Loop shaping) Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University