Time domain response specifications Dynamic Response Unit step signal: Step response: y(s)=H(s)/s, y(t)=L-1{H(s)/s} Time domain response specifications Defined based on unit step response Defined for closed-loop system

Transient Response First order system transient response Step response specs and relationship to pole location Second order system transient response Effects of additional poles and zeros

Prototype first order system 1 τs Y(s) U(s) + -

First order system step resp Normalized time t/t

Prototype first order system No overshoot, tp=inf, Mp = 0 Yss=1, ess=0 Settling time ts = [-ln(tol)]/p Delay time td = [-ln(0.5)]/p Rise time tr = [ln(0.9) – ln(0.1)]/p All times proportional to 1/p= t Larger p means faster response

The error signal: e(t) = 1-y(t)=e-ptus(t) Normalized time t/t

In every τ seconds, the error is reduced by 63.2%

General First-order system We know how this responds to input Step response starts at y(0+)=k, final value kz/p 1/p = t is still time constant; in every t, y(t) moves 63.2% closer to final value

Unit ramp response:

Note: In step response, the steady-state tracking error = zero.

Unit impulse response:

Prototype 2nd order system:

xi=[0.7 1 2 5 10 0.1 0.2 0.3 0.4 0.5 0.6]; x=['\zeta=0.7'; '\zeta=1 '; '\zeta=2 '; '\zeta=5 '; '\zeta=10 '; '\zeta=0.1'; '\zeta=0.2'; '\zeta=0.3'; '\zeta=0.4'; '\zeta=0.5'; '\zeta=0.6']; T=0:0.01:16; figure; hold; for k=1:length(xi) n=[1]; d=[1 2*xi(k) 1]; y=step(n,d,T); plot(T,y); if xi(k)>=0.7 text(T(290),y(310),x(k,:)); else text(T(290),max(y)+0.02,x(k,:)); end grid; text(9,1.65,'G(s)=w_n^2/(s^2+2\zetaw_ns+w_n^2)') title('Unit step responses for various \zeta') xlabel('w_nt (radians)') Can use \omega in stead of w

annotation Create annotations including lines, arrows, text arrows, double arrows, text boxes, rectangles, and ellipses xlabel, ylabel, zlabel Add a text label to the respective axis title Add a title to a graph colorbar Add a colorbar to a graph legend Add a legend to a graph

For example: “help annotation” explains how to use the annotation command to add text, lines, arrows, and so on at desired positions in the graph ANNOTATION('textbox',POSITION) creates a textbox annotation at the position specified in normalized figure units by the vector POSITION ANNOTATION('line',X,Y) creates a line annotation with endpoints specified in normalized figure coordinates by the vectors X and Y ANNOTATION('arrow',X,Y) creates an arrow annotation with endpoints specified Example: ah=annotation('arrow',[.9 .5],[.9,.5],'Color','r'); th=annotation('textarrow',[.3,.6],[.7,.4],'String','ABC');

Unit step response: 1) Under damped, 0 < ζ < 1

d s =Im =-Re cosq = z =-Re/|root| = cos-1(Re/|root|) = tan-1(-Re/Im)

To find y(t) max:

z=0.3:0.1:0.8; Mp=exp(-pi*z./sqrt(1-z.*z))*100 plot(z,Mp) grid; Then preference -> figure… ->powerpoint -> apply to figure Then copy figure

For 5% tolerance Ts ~= 3/zwn

Delay time is not used very much For delay time, solve y(t)=0.5 and solve for t For rise time, set y(t) = 0.1 & 0.9, solve for t This is very difficult Based on numerical simulation:

Useful Range td=(0.8+0.9z)/wn

Useful Range tr=4.5(z-0.2)/wn Or about 2/wn

Putting all things together: Settling time: