ME45: Control Systems Lecture Time response of nd-order systems Prof. Clar Radcliffe and Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Laplace transform Transfer function Models for systems electrical mechanical electromechanical Bloc diagrams Linearization Course roadmap Analysis Time response Transient Steady state Frequency response Bode plot Stability Routh-Hurwitz Nyquist Matlab simulations &) laboratories Design Design specs Root locus Frequency domain PID & Lead-lag Design examples nd Order Systems th Order Systems: parameter st Order Systems: parameters Time to respond called Time Constant nd Order Systems: 3 parameters Time to respond called Time Constant Oscillation rate called Frequency 3 Second-order systems A standard form of the second-order system! # n " : damping ratio s $ + "! n )s +! n! n : undamped natural frequency & Note: the boo does not include gain in its standard form DC motor position control example Amplifier Motor Closed-loop TF Step response for nd-order system Input a unit step function to a nd-order system. What is the output? ut) DC gain for a unit step if Gs) is stable) lim t!" s + "! n )s +! n # ) = lim sgs)us) s! ) = lim sgs) & s! $ s = lim Gs) s! ) = G) = Step response for nd-order system for various damping ratio.5.5 5 5 4 5 6
Step response for nd-order system Underdamped case Math expression of for underdamped case #! & Y s) = Gs)Us) = n s + $ "! n )s +! n s =!! " sin # t + $ ) $ " # d! = tan " & ) & # Damped natural frequency 7 Step response for nd-order system Underdamped case Math expression of for underdamped case =!! " sin # t + $ ) d & Damped response frequency sponse time constant! = "# n $ " #! = tan " & ) # 8 Transfer Function Poles For the nd order transfer function s + "! n )s +! n The denominator ==> Characteristic Polynomial s +!" n )s + " n The Characteristic Polynomial roots are the poles nd order ==> roots ==> two poles s,! " 9 Transfer Function Poles Poles for the nd order transfer function s s + "! n )s +!,! " n Step sponse =!! " sin # t + $ ) d &! = "# n The system poles set the response characteristics s, ) =/τ and s, ) = ω d Step response for nd-order system.5.5 5 5 Complex conjugate imaginary poles s,! " =! Step response for nd-order system.5.5 5 5 Complex conjugate complex poles s,! " =
Step response for nd-order system.5.5 5 5 Step response for nd-order system.5.5 5 5 nd Order System Poles Pole positions in the complex plane govern system response s) overdamped underdamped Increasing ζ s peated negative-real poles s,! " =!# n Two al poles s,! " =!" ± "! )# n s) underdamped Increasing ζ undamped 3 4 5 Poles <ζ<) <) Pole locations of G Pea Value Underdamped Underdamped) Assume!! " # $ $ Pea sponse M pt = &! e &! "#!" ) ) =!! " sin # t + $ ) d & P.O. vs. damping ratio Damping ratio overshoot = M pt! c ss c ss x Next, we clarify the influence of pole location on step response. 6 = e!"#!" x Pea sponse Overshoot Pea time! =. pea time T P =!! = " d " n # $ 7 8 3
Settling Time Measures how long it taes to stay within a set percentage of steady state Example =. =!! " sin #. = e!"# nt = e!t $ dt + $ & t $ =! ln.) = 3.9 4 So a system settles within of final value in about t 4τ Properties of nd-order system Influence of real part of poles Settling time ts decreases. settling time = 4τ 9 5) ) ts Influence of imag.. part of poles Oscillation frequency ω increases. d Influence of angle of poles Over/under-shoot decreases. An example quire 5 settling time ts < tsm given): 3 4 4
An example cont d) quire PO < POm given): An example cont d) Combination of two requirements & Some remars Percent overshoot depends on ζ,, but NOT ωn. From nd-order transfer function, analytic expressions of delay & rise time are hard to obtain. Time constant is /ζω ζωn), indicating convergence speed. For ζ>, we cannot define pea time, pea value, percent overshoot. 5 6 7 Performance measures review) Transient response Pea value Pea time Percent overshoot Delay time Rise time Settling time Steady state response Steady state gain Today s lecture) Next, we will connect these measures with s-domain. Summary Transient response of nd-order system is characterized by Damping ratio ζ & undamped natural frequency ωn Pole locations Delay time and rise time are not so easy to characterize, and thus not covered in this course. For transient responses of high order systems, we need computer simulations. 8 9 5