Transient Response First order system transient response Step
![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_h2/66841ada8a5897a3690d6373960e9748/image-1.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_h2/66841ada8a5897a3690d6373960e9748/image-2.jpg)
![First order system step resp Normalized time t/t First order system step resp Normalized time t/t](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-3.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_h2/66841ada8a5897a3690d6373960e9748/image-4.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_h2/66841ada8a5897a3690d6373960e9748/image-5.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_h2/66841ada8a5897a3690d6373960e9748/image-6.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_h2/66841ada8a5897a3690d6373960e9748/image-7.jpg)
![Step response by MATLAB: >> p =. . >> n = [ b 1 Step response by MATLAB: >> p =. . >> n = [ b 1](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-8.jpg)
![Unit ramp response: Unit ramp response:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-9.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-10.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_h2/66841ada8a5897a3690d6373960e9748/image-11.jpg)
![Unit impulse response: Unit impulse response:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-12.jpg)
![Prototype nd 2 order system: Prototype nd 2 order system:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-13.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-14.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-15.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_h2/66841ada8a5897a3690d6373960e9748/image-16.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_h2/66841ada8a5897a3690d6373960e9748/image-17.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_h2/66841ada8a5897a3690d6373960e9748/image-18.jpg)
![Unit step response: 1) Under damped, 0 < ζ < 1 Unit step response: 1) Under damped, 0 < ζ < 1](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-19.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_h2/66841ada8a5897a3690d6373960e9748/image-20.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-21.jpg)
![To find y(t) max: To find y(t) max:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-22.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-23.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-24.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_h2/66841ada8a5897a3690d6373960e9748/image-25.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-26.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-27.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-28.jpg)
![For 5% tolerance Ts ~= 3/zwn For 5% tolerance Ts ~= 3/zwn](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-29.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_h2/66841ada8a5897a3690d6373960e9748/image-30.jpg)
![Useful Range Td=(0. 8+0. 9 z)/wn Useful Range Td=(0. 8+0. 9 z)/wn](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-31.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_h2/66841ada8a5897a3690d6373960e9748/image-32.jpg)
![Putting all things together: Settling time: Putting all things together: Settling time:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-33.jpg)
![2) When ζ = 1, ωd = 0 2) When ζ = 1, ωd = 0](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-34.jpg)
![The tracking error: The tracking error:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-35.jpg)
![3) Over damped: ζ > 1 3) Over damped: ζ > 1](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-36.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-37.jpg)
![Transient Response Recall 1 st order system step response: 2 nd order: Transient Response Recall 1 st order system step response: 2 nd order:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-38.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-39.jpg)
![Pole location determines transient Pole location determines transient](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-40.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-41.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-42.jpg)
![• All closed-loop poles must be strictly in the left half planes Transient • All closed-loop poles must be strictly in the left half planes Transient](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-43.jpg)
![Typical design specifications • Steady-state: ess to step ≤ # % ts ≤ · Typical design specifications • Steady-state: ess to step ≤ # % ts ≤ ·](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-44.jpg)
![These specs translate into requirements on ζ, ωn or on closed-loop pole location : These specs translate into requirements on ζ, ωn or on closed-loop pole location :](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-45.jpg)
![Find conditions on σ and ωd. Find conditions on σ and ωd.](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-46.jpg)
![In the complex plane : In the complex plane :](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-47.jpg)
![Constant σ : vertical lines σ > # is half plane Constant σ : vertical lines σ > # is half plane](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-48.jpg)
![Constant ωd : horizontal line ωd < · · · is a band ωd Constant ωd : horizontal line ωd < · · · is a band ωd](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-49.jpg)
![Constant ωn : circles ωn < · · · inside of a circle ωn Constant ωn : circles ωn < · · · inside of a circle ωn](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-50.jpg)
![Constant ζ : φ = cos-1ζ constant Constant ζ = ray from the origin Constant ζ : φ = cos-1ζ constant Constant ζ = ray from the origin](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-51.jpg)
![If more than one requirement, get the common (overlapped) area e. g. ζ > If more than one requirement, get the common (overlapped) area e. g. ζ >](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-52.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-53.jpg)
![Try to remember these: Try to remember these:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-54.jpg)
![Example: + - When given unit step input, the output looks like: Q: estimate Example: + - When given unit step input, the output looks like: Q: estimate](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-55.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-56.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-57.jpg)
![Effects of additional zeros Suppose we originally have: i. e. step response Now introduce Effects of additional zeros Suppose we originally have: i. e. step response Now introduce](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-58.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-59.jpg)
![Effects: • Increased speed, • Larger overshoot, • Might increase ts Effects: • Increased speed, • Larger overshoot, • Might increase ts](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-60.jpg)
![When z < 0, the zero s = -z is > 0, is in When z < 0, the zero s = -z is > 0, is in](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-61.jpg)
![Effects of additional pole Suppose, instead of a zero, we introduce a pole at Effects of additional pole Suppose, instead of a zero, we introduce a pole at](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-62.jpg)
![L. P. F. has smoothing effect, or averaging effect Effects: • Slower, • Reduced L. P. F. has smoothing effect, or averaging effect Effects: • Slower, • Reduced](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-63.jpg)
- Slides: 63
![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_h2/66841ada8a5897a3690d6373960e9748/image-1.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_h2/66841ada8a5897a3690d6373960e9748/image-2.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_h2/66841ada8a5897a3690d6373960e9748/image-3.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_h2/66841ada8a5897a3690d6373960e9748/image-4.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_h2/66841ada8a5897a3690d6373960e9748/image-5.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_h2/66841ada8a5897a3690d6373960e9748/image-6.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_h2/66841ada8a5897a3690d6373960e9748/image-7.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
![Step response by MATLAB p n b 1 Step response by MATLAB: >> p =. . >> n = [ b 1](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-8.jpg)
Step response by MATLAB: >> p =. . >> n = [ b 1 b 0 ] >> 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 Unit ramp response:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-9.jpg)
Unit ramp response:
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-10.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_h2/66841ada8a5897a3690d6373960e9748/image-11.jpg)
Note: In step response, the steady-state tracking error = zero.
![Unit impulse response Unit impulse response:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-12.jpg)
Unit impulse response:
![Prototype nd 2 order system Prototype nd 2 order system:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-13.jpg)
Prototype nd 2 order system:
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-14.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-15.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_h2/66841ada8a5897a3690d6373960e9748/image-16.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_h2/66841ada8a5897a3690d6373960e9748/image-17.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_h2/66841ada8a5897a3690d6373960e9748/image-18.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_h2/66841ada8a5897a3690d6373960e9748/image-19.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_h2/66841ada8a5897a3690d6373960e9748/image-20.jpg)
d =Im cosq = z =-Re/|root| q= cos-1(Re/|root|) q= tan-1(-Re/Im) s =-Re
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-21.jpg)
![To find yt max To find y(t) max:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-22.jpg)
To find y(t) max:
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-23.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-24.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_h2/66841ada8a5897a3690d6373960e9748/image-25.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_h2/66841ada8a5897a3690d6373960e9748/image-26.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-27.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-28.jpg)
![For 5 tolerance Ts 3zwn For 5% tolerance Ts ~= 3/zwn](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-29.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_h2/66841ada8a5897a3690d6373960e9748/image-30.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_h2/66841ada8a5897a3690d6373960e9748/image-31.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_h2/66841ada8a5897a3690d6373960e9748/image-32.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_h2/66841ada8a5897a3690d6373960e9748/image-33.jpg)
Putting all things together: Settling time:
![2 When ζ 1 ωd 0 2) When ζ = 1, ωd = 0](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-34.jpg)
2) When ζ = 1, ωd = 0
![The tracking error The tracking error:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-35.jpg)
The tracking error:
![3 Over damped ζ 1 3) Over damped: ζ > 1](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-36.jpg)
3) Over damped: ζ > 1
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-37.jpg)
![Transient Response Recall 1 st order system step response 2 nd order Transient Response Recall 1 st order system step response: 2 nd order:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-38.jpg)
Transient Response Recall 1 st order system step response: 2 nd order:
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-39.jpg)
![Pole location determines transient Pole location determines transient](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-40.jpg)
Pole location determines transient
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-41.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-42.jpg)
![All closedloop poles must be strictly in the left half planes Transient • All closed-loop poles must be strictly in the left half planes Transient](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-43.jpg)
• 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 Steadystate ess to step ts Typical design specifications • Steady-state: ess to step ≤ # % ts ≤ ·](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-44.jpg)
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 closedloop pole location These specs translate into requirements on ζ, ωn or on closed-loop pole location :](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-45.jpg)
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 Find conditions on σ and ωd.](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-46.jpg)
Find conditions on σ and ωd.
![In the complex plane In the complex plane :](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-47.jpg)
In the complex plane :
![Constant σ vertical lines σ is half plane Constant σ : vertical lines σ > # is half plane](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-48.jpg)
Constant σ : vertical lines σ > # is half plane
![Constant ωd horizontal line ωd is a band ωd Constant ωd : horizontal line ωd < · · · is a band ωd](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-49.jpg)
Constant ωd : horizontal line ωd < · · · is a band ωd > · · · is the plane excluding band
![Constant ωn circles ωn inside of a circle ωn Constant ωn : circles ωn < · · · inside of a circle ωn](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-50.jpg)
Constant ωn : circles ωn < · · · inside of a circle ωn > · · · outside of a circle
![Constant ζ φ cos1ζ constant Constant ζ ray from the origin Constant ζ : φ = cos-1ζ constant Constant ζ = ray from the origin](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-51.jpg)
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 If more than one requirement, get the common (overlapped) area e. g. ζ >](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-52.jpg)
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.
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-53.jpg)
![Try to remember these Try to remember these:](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-54.jpg)
Try to remember these:
![Example When given unit step input the output looks like Q estimate Example: + - When given unit step input, the output looks like: Q: estimate](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-55.jpg)
Example: + - When given unit step input, the output looks like: Q: estimate k and τ.
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-56.jpg)
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-57.jpg)
![Effects of additional zeros Suppose we originally have i e step response Now introduce Effects of additional zeros Suppose we originally have: i. e. step response Now introduce](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-58.jpg)
Effects of additional zeros Suppose we originally have: i. e. step response Now introduce a zero at s = -z The new step response:
![](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-59.jpg)
![Effects Increased speed Larger overshoot Might increase ts Effects: • Increased speed, • Larger overshoot, • Might increase ts](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-60.jpg)
Effects: • Increased speed, • Larger overshoot, • Might increase ts
![When z 0 the zero s z is 0 is in When z < 0, the zero s = -z is > 0, is in](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-61.jpg)
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 Effects of additional pole Suppose, instead of a zero, we introduce a pole at](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-62.jpg)
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 L. P. F. has smoothing effect, or averaging effect Effects: • Slower, • Reduced](https://slidetodoc.com/presentation_image_h2/66841ada8a5897a3690d6373960e9748/image-63.jpg)
L. P. F. has smoothing effect, or averaging effect Effects: • Slower, • Reduced overshoot, • May increase or decrease ts
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