Dynamic Response Unit step signal Step response ysHss
![Dynamic Response • Unit step signal: • Step response: y(s)=H(s)/s, y(t)=L-1{H(s)/s} Time domain response Dynamic Response • Unit step signal: • Step response: y(s)=H(s)/s, y(t)=L-1{H(s)/s} Time domain response](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-1.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-2.jpg)
![Transient Response • First order system transient response – Step response specs and relationship Transient Response • First order system transient response – Step response specs and relationship](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-3.jpg)
![Prototype first order system E U(s) + - 1 τs Y(s) Prototype first order system E U(s) + - 1 τs Y(s)](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-4.jpg)
![First order system step resp Normalized time t/t First order system step resp Normalized time t/t](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-5.jpg)
![Prototype first order system • • • No overshoot, tp=inf, Mp = 0 Yss=1, Prototype first order system • • • No overshoot, tp=inf, Mp = 0 Yss=1,](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-6.jpg)
![The error signal: e(t) = 1 -y(t)=e-ptus(t) Normalized time t/t The error signal: e(t) = 1 -y(t)=e-ptus(t) Normalized time t/t](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-7.jpg)
![In every τ seconds, the error is reduced by 63. 2% In every τ seconds, the error is reduced by 63. 2%](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-8.jpg)
![General First-order system We know how this responds to input Step response starts at General First-order system We know how this responds to input Step response starts at](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-9.jpg)
![Unit ramp response: Unit ramp response:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-10.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-11.jpg)
![Note: In step response, the steady-state tracking error = zero. Note: In step response, the steady-state tracking error = zero.](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-12.jpg)
![Unit impulse response: Unit impulse response:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-13.jpg)
![Prototype nd 2 order system: Prototype nd 2 order system:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-14.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-15.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-16.jpg)
![xi=[0. 7 1 2 5 10 0. 1 0. 2 0. 3 0. 4 xi=[0. 7 1 2 5 10 0. 1 0. 2 0. 3 0. 4](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-17.jpg)
![annotation Create annotations including lines, arrows, text arrows, double arrows, text boxes, rectangles, and annotation Create annotations including lines, arrows, text arrows, double arrows, text boxes, rectangles, and](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-18.jpg)
![For example: “help annotation” explains how to use the annotation command to add text, For example: “help annotation” explains how to use the annotation command to add text,](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-19.jpg)
![Unit step response: 1) Under damped, 0 < ζ < 1 Unit step response: 1) Under damped, 0 < ζ < 1](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-20.jpg)
![d =Im cosq = z =-Re/|root| q= cos-1(Re/|root|) q= tan-1(-Re/Im) s =-Re d =Im cosq = z =-Re/|root| q= cos-1(Re/|root|) q= tan-1(-Re/Im) s =-Re](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-21.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-22.jpg)
![To find y(t) max: To find y(t) max:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-23.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-24.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-25.jpg)
![z=0. 3: 0. 1: 0. 8; Mp=exp(-pi*z. /sqrt(1 -z. *z))*100 plot(z, Mp) grid; Then z=0. 3: 0. 1: 0. 8; Mp=exp(-pi*z. /sqrt(1 -z. *z))*100 plot(z, Mp) grid; Then](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-26.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-27.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-28.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-29.jpg)
![For 5% tolerance Ts ~= 3/zwn For 5% tolerance Ts ~= 3/zwn](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-30.jpg)
![• Delay time is not used very much • For delay time, solve • Delay time is not used very much • For delay time, solve](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-31.jpg)
![Useful Range td=(0. 8+0. 9 z)/wn Useful Range td=(0. 8+0. 9 z)/wn](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-32.jpg)
![Useful Range tr=4. 5(z-0. 2)/wn Or about 2/wn Useful Range tr=4. 5(z-0. 2)/wn Or about 2/wn](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-33.jpg)
![Putting all things together: Settling time: Putting all things together: Settling time:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-34.jpg)
- Slides: 34
![Dynamic Response Unit step signal Step response ysHss ytL1Hss Time domain response Dynamic Response • Unit step signal: • Step response: y(s)=H(s)/s, y(t)=L-1{H(s)/s} Time domain response](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-1.jpg)
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
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-2.jpg)
![Transient Response First order system transient response Step response specs and relationship Transient Response • First order system transient response – Step response specs and relationship](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-3.jpg)
Transient Response • First order system transient response – Step response specs and relationship to pole location • Second order system transient response – Step response specs and relationship to pole location • Effects of additional poles and zeros
![Prototype first order system E Us 1 τs Ys Prototype first order system E U(s) + - 1 τs Y(s)](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-4.jpg)
Prototype first order system E U(s) + - 1 τs Y(s)
![First order system step resp Normalized time tt First order system step resp Normalized time t/t](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-5.jpg)
First order system step resp Normalized time t/t
![Prototype first order system No overshoot tpinf Mp 0 Yss1 Prototype first order system • • • No overshoot, tp=inf, Mp = 0 Yss=1,](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-6.jpg)
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 et 1 yteptust Normalized time tt The error signal: e(t) = 1 -y(t)=e-ptus(t) Normalized time t/t](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-7.jpg)
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 In every τ seconds, the error is reduced by 63. 2%](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-8.jpg)
In every τ seconds, the error is reduced by 63. 2%
![General Firstorder system We know how this responds to input Step response starts at General First-order system We know how this responds to input Step response starts at](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-9.jpg)
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 Unit ramp response:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-10.jpg)
Unit ramp response:
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-11.jpg)
![Note In step response the steadystate tracking error zero Note: In step response, the steady-state tracking error = zero.](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-12.jpg)
Note: In step response, the steady-state tracking error = zero.
![Unit impulse response Unit impulse response:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-13.jpg)
Unit impulse response:
![Prototype nd 2 order system Prototype nd 2 order system:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-14.jpg)
Prototype nd 2 order system:
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-15.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-16.jpg)
![xi0 7 1 2 5 10 0 1 0 2 0 3 0 4 xi=[0. 7 1 2 5 10 0. 1 0. 2 0. 3 0. 4](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-17.jpg)
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; end text(9, 1. 65, 'G(s)=w_n^2/(s^2+2zetaw_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 annotation Create annotations including lines, arrows, text arrows, double arrows, text boxes, rectangles, and](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-18.jpg)
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 For example: “help annotation” explains how to use the annotation command to add text,](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-19.jpg)
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 Unit step response: 1) Under damped, 0 < ζ < 1](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-20.jpg)
Unit step response: 1) Under damped, 0 < ζ < 1
![d Im cosq z Reroot q cos1Reroot q tan1ReIm s Re d =Im cosq = z =-Re/|root| q= cos-1(Re/|root|) q= tan-1(-Re/Im) s =-Re](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-21.jpg)
d =Im cosq = z =-Re/|root| q= cos-1(Re/|root|) q= tan-1(-Re/Im) s =-Re
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-22.jpg)
![To find yt max To find y(t) max:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-23.jpg)
To find y(t) max:
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-24.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-25.jpg)
![z0 3 0 1 0 8 Mpexppiz sqrt1 z z100 plotz Mp grid Then z=0. 3: 0. 1: 0. 8; Mp=exp(-pi*z. /sqrt(1 -z. *z))*100 plot(z, Mp) grid; Then](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-26.jpg)
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
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-27.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-28.jpg)
![](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-29.jpg)
![For 5 tolerance Ts 3zwn For 5% tolerance Ts ~= 3/zwn](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-30.jpg)
For 5% tolerance Ts ~= 3/zwn
![Delay time is not used very much For delay time solve • Delay time is not used very much • For delay time, solve](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-31.jpg)
• 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 td0 80 9 zwn Useful Range td=(0. 8+0. 9 z)/wn](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-32.jpg)
Useful Range td=(0. 8+0. 9 z)/wn
![Useful Range tr4 5z0 2wn Or about 2wn Useful Range tr=4. 5(z-0. 2)/wn Or about 2/wn](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-33.jpg)
Useful Range tr=4. 5(z-0. 2)/wn Or about 2/wn
![Putting all things together Settling time Putting all things together: Settling time:](https://slidetodoc.com/presentation_image/1bc814a6ed9ef1905a24e4eef4dfac1e/image-34.jpg)
Putting all things together: Settling time:
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