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%

We know how this responds to input 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

Step response by MATLAB: >> n = [ b1 b0 ] >> d = [ 1 p ] >> step ( n , d ) Other MATLAB commands to explore: plot, hold, axis, xlabel, ylabel, title, text, gtext, semilogx, semilogy, loglog, subplot

Unit ramp response:

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

Unit impulse response:

Prototype 2nd order system:

Can use \omega in stead of w 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

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

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:

2) When ζ = 1, ωd = 0

The tracking error:

3) Over damped: ζ > 1

Transient Response Recall 1st order system step response: 2nd order:

Pole location determines transient

All closed-loop poles must be strictly in the left half planes Transient dies away Dominant poles: the single real pole or the complex pole pair which contribute the most to the transient Typically have dominant pole pair (complex conjugate) Closest to jω-axis (i.e. the least negative) Slowest to die away

Typical design specifications Steady-state: ess to step ≤ # % ts ≤ · · · Speed (responsiveness) tr ≤ · · · td ≤ · · · Relative stability Mp ≤ · · · %

These specs translate into requirements on ζ, ωn or on closed-loop pole location : Find ranges for ζ and ωn so that all 3 are satisfied.

Find conditions on σ and ωd.

In the complex plane :

Constant σ : vertical lines σ > # is half plane

Constant ωd : horizontal line ωd < · · · is a band ωd > · · · is the plane excluding band

Constant ωn : circles ωn < · · · inside of a circle ωn > · · · outside of a circle

Constant ζ : φ = cos-1ζ constant Constant ζ = ray from the origin ζ > · · · is the cone ζ < · · · is the other part

If more than one requirement, get the common (overlapped) area e.g. ζ > 0.5, σ > 2, ωn > 3 gives Sometimes meeting two will also meet the third, but not always.

Try to remember these:

When given unit step input, the output looks like: Example: + - When given unit step input, the output looks like: Q: estimate k and τ.

Effects of additional zeros Suppose we originally have: i.e. step response Now introduce a zero at s = -z The new step response:

Effects: Increased speed, Larger overshoot, Might increase ts

When z < 0, the zero s = -z is > 0, is in the right half plane. Such a zero is called a nonminimum phase zero. A system with nonminimum phase zeros is called a nonminimum phase system. Nonminimum phase zero should be avoided in design. i.e. Do not introduce such a zero in your controller.

Effects of additional pole Suppose, instead of a zero, we introduce a pole at s = -p, i.e.

L.P.F. has smoothing effect, or averaging effect Effects: Slower, Reduced overshoot, May increase or decrease ts